#7596948

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2105 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

(2,3)(5,6)(7,11)(8,9)(10,12)(13,14,17,19)(15,21,25,28)(16,26,22,20)(18,27,23,24);(2,3)(5,6)(7,10)(8,9)(11,12)(13,15,22,27,17,25,16,24)(14,18,26,28,19,23,20,21);(1,2,4,3)(6,9)(10,11)(13,14)(15,23)(16,20)(17,19)(18,25)(21,27)(22,26)(24,28);(1,3,4,2)(5,7,8,12)(6,10,9,11)(14,19)(18,23)(20,26)(21,28);(13,15,22,27,17,25,16,24)(14,18,26,28,19,23,20,21)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=h,d^2=gi,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7504144);