#7622659

order	512
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₂⁴
commutator, socle	C₂²	C₂⁴×C₈
Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	#3455	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1358 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=f,g^2=h,h^2=i,b^a=bf,e^a=ei,d^b=di

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529855);