#6559056

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535823
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(2,4)(5,6)(7,15)(8,17)(9,13)(10,12)(11,14)(16,19)(18,20)(21,22)(23,25)(24,32)(26,31)(27,29)(28,34)(30,36)(33,35);(5,7,11,10)(6,12,14,15)(8,18,9,16)(13,20,17,19)(21,23,25,22,27,31,26,29)(24,33,35,32,34,30,36,28);(1,2,3,4)(5,8,11,9)(6,13,14,17)(7,16,10,18)(12,19,15,20)(21,24,27,34)(22,28,29,32)(23,30,31,33)(25,35,26,36);(5,9,11,8)(6,13,14,17)(7,16,10,18)(12,20,15,19)(24,34)(28,32)(30,33)(35,36);(5,10,11,7)(6,12,14,15)(8,18,9,16)(13,19,17,20)(21,25,27,26)(22,23,29,31)(24,35,34,36)(28,30,32,33)

Polycyclic presentation:

2^9|b^2=fi,c^2=g,d^2=hi,e^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ei,d^c=di,e^d=ehi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307435, #72307590, #72307872, …
maximal normal subgroup of	#72307435, #72307590, #72307872, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6466252);