#7500808

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	C₂×#3002	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3256 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)(9,10,15,18,14,20,13,19)(11,21,23,16,25,35,24,32)(12,26,28,17,30,36,29,34)(22,31,39,37,38,40,27,33);(5,6)(7,8)(9,11)(10,16)(12,27)(13,23)(14,25)(15,24)(17,33)(18,21)(19,35)(20,32)(22,29)(26,31)(28,38)(30,39)(34,37)(36,40);(2,4)(6,8);(2,4)(9,12)(10,17)(11,22)(13,28)(14,30)(15,29)(16,31)(18,26)(19,36)(20,34)(21,37)(23,27)(24,39)(25,38)(32,40)(33,35);(9,13,14,15)(10,18,20,19)(11,23,25,24)(12,28,30,29)(16,21,32,35)(17,26,34,36)(22,27,38,39)(31,37,40,33)

Polycyclic presentation:

2^9|a^2=fi,e^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, 7408004);