#7527774

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	#35771	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1995 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,h^b=hi,e^c=ei,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, 7434970);