#6377548

order	512
nilpotency class	3
isoclinism	isoclinic class of #60771
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2088 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cfi,g^c=gi,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, 6284744);