#7503492

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1856 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^c=ei,g^c=gi,g^d=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7410688);