#513302

order	512
nilpotency class	5
isoclinism	isoclinic class of #513173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	#13729	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ Q₈×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

332 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 72 points:

(2,6)(7,8)(9,10)(11,22)(12,17)(13,31)(14,35)(15,21)(16,25)(18,33)(19,29)(20,30)(23,49)(24,50)(26,56)(27,57)(28,58)(32,46)(34,64)(36,43)(37,42)(38,51)(39,53)(40,66)(41,60)(44,70)(45,67)(47,62)(48,55)(52,71)(54,69)(59,68)(61,63)(65,72);(1,2,5,6)(3,7,4,8)(11,23)(16,38)(22,49)(24,27)(25,51)(26,52)(32,65)(36,54)(39,48)(40,68)(43,69)(46,72)(50,57)(53,55)(56,71)(59,66);(1,3,5,4)(2,7,6,8)(9,11)(10,16)(12,24)(13,32)(14,36)(15,23)(17,39)(18,43)(19,46)(20,48)(21,38)(22,31)(25,33)(26,29)(27,30)(28,59)(34,65)(35,40)(37,54)(41,56)(42,68)(44,71)(45,69)(47,72)(49,64)(50,58)(51,67)(52,62)(53,60)(55,70)(57,63)(61,66);(1,4,5,3)(2,7,6,8)(9,12,15,30)(10,17,21,20)(11,24,23,27)(13,28,34,61)(14,29,37,62)(16,39,38,48)(18,41,45,44)(19,42,47,35)(22,50,49,57)(25,53,51,55)(26,54,52,36)(31,58,64,63)(32,59,65,66)(33,60,67,70)(40,46,68,72)(43,56,69,71)

Polycyclic presentation:

2^9|b^2=e,c^2=e,d^2=ei,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fg,g^a=gh,h^a=hi,c^b=ci,d^b=de,f^c=fg,g^c=gh,h^c=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234374, #72322380, #72322412
maximal normal subgroup of	#72234374, #72322380, #72322412

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420498);