#7501754

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 ◁ #2296 ◁ 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)(13,14,18,24)(15,26,30,20)(16,31,33,39)(17,25,19,23)(21,35,22,32)(27,41,28,44)(29,37,34,40)(36,43,38,42);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15)(14,20)(16,27)(17,34)(18,30)(19,29)(21,36)(22,38)(23,40)(24,26)(25,37)(28,33)(31,44)(32,43)(35,42)(39,41);(2,4)(5,8,9,12)(6,10,7,11)(13,16)(14,21)(15,27)(17,32)(18,33)(19,35)(20,36)(22,24)(23,31)(25,39)(26,38)(28,30)(29,42)(34,43)(37,41)(40,44);(6,7)(10,11)(13,16)(14,22)(15,28)(17,32)(18,33)(19,35)(20,36)(21,24)(23,39)(25,31)(26,38)(27,30)(29,43)(34,42)(37,41)(40,44);(13,17,18,19)(14,23,24,25)(15,29,30,34)(16,32,33,35)(20,37,26,40)(21,31,22,39)(27,42,28,43)(36,41,38,44)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fi,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^b=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, 7408950);