#6399072

order	512
nilpotency class	3
isoclinism	isoclinic class of #61609
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#555	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1810 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=fi,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,g^a=gi,c^b=ci,d^b=df,e^b=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6306268);