#7149133

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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

1904 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)(13,14,18,22,17,19,30,33)(15,23,26,32,25,37,31,20)(16,27,29,43,28,21,35,36)(24,39,41,34,40,38,44,42);(1,2,3,4)(6,9)(7,11)(14,19)(16,28)(20,32)(22,33)(23,37)(24,40)(29,35)(41,44);(5,6)(7,10)(8,9)(11,12)(13,15,17,25)(14,20,19,32)(16,24,28,40)(18,31,30,26)(21,34,27,42)(22,37,33,23)(29,44,35,41)(36,39,43,38);(2,4)(6,9)(10,12)(13,16,17,28)(14,21,19,27)(15,24,25,40)(18,29,30,35)(20,34,32,42)(22,36,33,43)(23,38,37,39)(26,41,31,44);(5,7,8,11)(6,10,9,12)(16,28)(21,27)(24,40)(29,35)(34,42)(36,43)(38,39)(41,44)

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,d^a=di,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

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7056329);