#7613842

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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 ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1984 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)(13,14,18,21)(15,17,23,19)(16,20,27,24)(22,28,25,26);(2,3)(6,9)(10,11)(13,15)(14,19)(16,24)(17,21)(18,23)(20,27)(22,26)(25,28);(1,2,4,3)(13,15)(14,17)(16,25)(18,23)(19,21)(20,26)(22,27)(24,28);(1,3,4,2)(13,16,19,26,18,27,17,28)(14,20,15,22,21,24,23,25);(5,7,8,12)(6,10,9,11)(13,17,18,19)(14,15,21,23)(16,26,27,28)(20,25,24,22)

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=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, 7521038);