#7527781

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	#36018	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2014 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)(15,24)(16,28)(17,32)(18,23)(19,22)(20,27)(21,31)(25,35)(26,36)(29,38)(30,39)(33,37)(34,40)(41,43)(42,44);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,15)(14,20)(16,29)(17,26)(18,27)(19,33)(21,38)(22,36)(23,24)(25,30)(28,44)(31,42)(32,37)(34,43)(35,40)(39,41);(6,10)(7,11)(13,15)(14,20)(16,25)(17,33)(18,27)(19,26)(21,35)(22,37)(23,24)(28,41)(29,30)(31,43)(32,36)(34,42)(38,40)(39,44);(6,10)(9,12)(13,16,18,31)(14,21,23,28)(15,25,27,43)(17,30,19,34)(20,35,24,41)(22,39,32,40)(26,42,33,29)(36,38,37,44);(1,3,4,2)(13,17,18,19)(14,22,23,32)(15,26,27,33)(16,30,31,34)(20,36,24,37)(21,39,28,40)(25,42,43,29)(35,38,41,44)

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,d^2=i,e^2=f,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, 7434977);