#7504127

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	#2414	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)(13,14,20,34,19,24,18,25)(15,21,35,44,28,39,29,38)(16,30,33,22,17,32,31,23)(26,40,43,36,27,42,41,37);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,19,28)(14,21,24,39)(16,26,17,27)(18,29,20,35)(22,36,23,37)(25,38,34,44)(30,40,32,42)(31,41,33,43);(2,4)(13,16,19,17)(14,22,24,23)(15,26,28,27)(18,33,20,31)(21,36,39,37)(25,32,34,30)(29,43,35,41)(38,42,44,40);(6,11)(7,12)(13,17,19,16)(14,23,24,22)(15,27,28,26)(18,31,20,33)(21,37,39,36)(25,30,34,32)(29,41,35,43)(38,40,44,42);(5,8,9,10)(6,7,11,12)(14,24)(15,28)(22,23)(25,34)(26,27)(29,35)(30,32)(41,43)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^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, 7411323);