#7501070

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2729 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=i,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=df,e^b=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, 7408266);