#7520158

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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	#25614	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1848 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)(7,11)(8,12)(9,10)(13,14,20,25)(15,21,29,42)(16,30,27,43)(17,34,26,44)(18,24,19,35)(22,32,39,33)(23,37,38,31)(28,40,36,41);(1,2)(3,4)(6,10)(11,12)(13,15,20,29)(14,21,25,42)(16,31,27,37)(17,32,26,33)(18,28,19,36)(22,34,39,44)(23,43,38,30)(24,40,35,41);(5,7,9,8)(6,11,10,12)(13,16,20,27)(14,22,25,39)(15,26,29,17)(18,33,19,32)(21,38,42,23)(24,43,35,30)(28,31,36,37)(34,40,44,41);(2,4)(13,17,20,26)(14,23,25,38)(15,27,29,16)(18,31,19,37)(21,39,42,22)(24,44,35,34)(28,33,36,32)(30,40,43,41);(5,8,9,7)(6,11,10,12)(13,18,20,19)(14,24,25,35)(15,28,29,36)(16,32,27,33)(17,31,26,37)(21,40,42,41)(22,30,39,43)(23,44,38,34)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,d^2=i,e^2=fi,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=cgi,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

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7427354);