#6403852

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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1813 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)(15,20)(16,26)(17,22)(18,32)(19,23)(21,27)(24,36)(25,33)(28,39)(29,34)(30,37)(31,42)(35,40)(38,44)(41,43);(5,7)(6,9)(8,12)(10,11)(13,15,19,18)(14,20,23,32)(16,27,29,24)(17,25,30,31)(21,34,36,26)(22,33,37,42)(28,40,41,38)(35,43,44,39);(6,10)(9,11)(13,16,18,27,19,29,15,24)(14,21,20,26,23,36,32,34)(17,28,31,40,30,41,25,38)(22,35,33,39,37,44,42,43);(7,12)(9,11)(13,17,19,30)(14,22,23,37)(15,25,18,31)(16,28,29,41)(20,33,32,42)(21,35,36,44)(24,38,27,40)(26,39,34,43);(1,3,4,2)(6,10)(9,11)(17,30)(22,37)(25,31)(28,41)(33,42)(35,44)(38,40)(39,43)

Polycyclic presentation:

2^9|a^2=h,b^2=i,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^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, 6311048);