#7500998

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×#3012	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3261 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)(9,10,13,18,14,19,27,17)(11,20,23,29,24,33,22,15)(12,16,26,32,25,31,38,30)(21,34,37,39,36,40,35,28);(5,6)(7,8)(9,11,14,24)(10,15,19,29)(12,21,25,36)(13,22,27,23)(16,28,31,39)(17,20,18,33)(26,35,38,37)(30,34,32,40);(2,4)(6,8)(9,12)(10,16)(11,21)(13,26)(14,25)(15,28)(17,30)(18,32)(19,31)(20,34)(22,35)(23,37)(24,36)(27,38)(29,39)(33,40);(2,4)(10,17)(11,22)(13,27)(16,30)(18,19)(20,33)(21,35)(23,24)(26,38)(31,32)(34,40)(36,37);(12,25)(16,31)(21,36)(26,38)(28,39)(30,32)(34,40)(35,37)

Polycyclic presentation:

2^9|a^2=f,b^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,f^b=fi,e^c=ei,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, 7408194);