#7515018

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1948 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)(7,10)(8,9)(11,12)(13,14,18,22)(15,24,27,20)(16,29,21,38)(17,32,34,31)(19,36,35,23)(25,44,39,33)(26,30,43,41)(28,40,37,42);(1,2)(3,4)(6,9)(10,12)(13,15,18,27)(14,20,22,24)(16,30,21,41)(17,33,34,44)(19,37,35,28)(23,42,36,40)(25,32,39,31)(26,38,43,29);(5,7,8,11)(6,10,9,12)(13,16)(14,17)(15,25)(18,21)(19,32)(20,26)(22,34)(23,38)(24,43)(27,39)(28,41)(29,36)(30,37)(31,35)(33,42)(40,44);(2,4)(13,17,18,34)(14,21,22,16)(15,26,27,43)(19,38,35,29)(20,39,24,25)(23,31,36,32)(28,44,37,33)(30,42,41,40);(6,9)(10,12)(14,22)(16,21)(20,24)(23,36)(25,39)(30,41)(31,32)(40,42)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,d^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7422214);