#7503340

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35659	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3552 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

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, 7410536);