#6399385

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1729 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)(22,28)(23,31)(25,27)(29,36)(32,33)(34,35);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,22,26,28)(23,32,35,29)(24,27,30,25)(31,36,34,33);(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(21,23,26,35)(22,29,28,32)(24,33,30,36)(25,31,27,34);(7,15)(10,13)(14,20)(17,19)(21,24,26,30)(22,25,28,27)(23,33,35,36)(29,31,32,34);(1,3,4,2)(6,12)(10,13)(11,18)(17,19)(21,25,26,27)(22,30,28,24)(23,34,35,31)(29,33,32,36)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=f,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,e^b=ei,e^c=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, 6306581);