#7502338

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500701
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2219 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,g^a=gi,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, 7409534);