#7504123

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1860 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,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, 7411319);