#6403908

order	512
nilpotency class	3
isoclinism	isoclinic class of #62283
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

1809 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2,4,3)(5,6)(7,11)(8,10)(9,12)(13,14)(15,24)(16,27)(17,20)(18,22)(19,33)(21,31)(23,29)(25,39)(26,35)(28,41)(30,42)(32,38)(34,43)(36,40)(37,44);(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,28)(17,18)(19,26)(21,36)(22,24)(23,35)(25,30)(27,39)(29,32)(31,43)(33,38)(34,37)(40,44)(41,42);(6,10)(9,11)(13,16,29,44,18,30,19,31)(14,21,33,42,22,37,23,27)(15,25,32,43,17,28,26,40)(20,34,38,39,24,36,35,41);(7,12)(9,11)(13,17,18,15)(14,20,22,24)(16,28,30,25)(19,32,29,26)(21,34,37,36)(23,35,33,38)(27,41,42,39)(31,43,44,40);(1,3,4,2)(6,10)(9,11)(14,22)(15,17)(21,37)(23,33)(25,28)(26,32)(27,42)(40,43)

Polycyclic presentation:

2^9|a^2=h,c^2=fg,d^2=i,e^2=h,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^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6311104);