#6526912

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500456
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
prime radical	#28316	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2048 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)(5,6,9,12)(7,10,13,16)(8,11,15,18)(14,17,19,20)(21,22,26,27)(23,30,29,25)(24,28,32,34)(31,36,35,33);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,25,22,26,29,30,27)(24,31,33,28,32,35,36,34);(1,2,3,4)(7,13)(8,15)(10,16)(11,18)(22,27)(23,29)(28,34)(31,35);(5,8,9,15)(6,11,12,18)(7,14,13,19)(10,17,16,20)(21,24,26,32)(22,28,27,34)(23,31,29,35)(25,33,30,36);(7,13)(10,16)(14,19)(17,20)(24,32)(28,34)(31,35)(33,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6434108);