#489970

order	512
nilpotency class	4
isoclinism	isoclinic class of #489719
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#121	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

699 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)(3,4)(7,8)(9,10,15,20)(11,21,23,49)(12,17,26,40)(13,29,33,47)(14,34,31,19)(16,36,38,24)(18,43,45,32)(22,48,52,70)(25,53,35,44)(27,56,54,69)(28,59,55,42)(30,46,61,39)(37,62,64,50)(41,66,65,58)(51,71,60,67)(57,68,72,63);(1,2)(3,6)(4,8)(5,7)(9,11,15,23)(10,16,20,38)(12,22,26,52)(13,30,33,61)(14,35,31,25)(17,37,40,64)(18,44,45,53)(19,46,34,39)(21,47,49,29)(24,43,36,32)(27,57,54,72)(28,60,55,51)(41,67,65,71)(42,68,59,63)(48,69,70,56)(50,66,62,58);(1,2)(3,6)(4,8)(5,7)(9,12)(10,17)(11,22)(13,27)(14,28)(15,26)(16,37)(18,41)(19,42)(20,40)(21,48)(23,52)(24,50)(25,51)(29,56)(30,57)(31,55)(32,58)(33,54)(34,59)(35,60)(36,62)(38,64)(39,63)(43,66)(44,67)(45,65)(46,68)(47,69)(49,70)(53,71)(61,72);(1,3,5,4)(2,6,7,8)(11,23)(12,26)(16,38)(17,40)(21,49)(24,36)(25,35)(27,54)(28,55)(30,61)(39,46)(41,65)(42,59)(44,53)(56,69)(58,66)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=h,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,g^a=gi,d^b=di,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397166);