#7149101

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1916 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=i,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

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, 7056297);