#7501804

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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	#23426	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2423 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)(7,9)(8,10)(11,12)(13,14,18,22)(15,19,25,36)(16,27,29,39)(17,30,33,23)(20,31,35,26)(21,32,38,28)(24,37,40,44)(34,42,43,41);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15)(14,19)(16,24)(17,26)(18,25)(20,30)(21,34)(22,36)(23,35)(27,37)(28,41)(29,40)(31,33)(32,42)(38,43)(39,44);(2,4)(5,8,9,12)(6,10,7,11)(13,15)(14,20)(16,24)(17,31)(18,25)(19,30)(21,37)(22,35)(23,36)(26,33)(27,34)(28,42)(29,40)(32,41)(38,44)(39,43);(6,7)(10,11)(13,16,18,29)(14,21,22,38)(15,24,25,40)(17,32,33,28)(19,34,36,43)(20,37,35,44)(23,39,30,27)(26,42,31,41);(14,22)(15,25)(21,38)(23,30)(24,40)(26,31)(27,39)(41,42)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^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^b=ei,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, 7409000);