#7604220

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
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1761 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)(5,6,9,8)(7,10,12,11)(13,14)(15,19)(16,23)(17,18)(20,22)(21,26)(24,27)(25,28);(1,2,4,3)(5,7)(6,10)(8,11)(9,12)(13,15,20,17)(14,19,22,18)(16,24,25,21)(23,27,28,26);(5,8,9,6)(7,11,12,10)(13,16,18,26,20,25,19,27)(14,21,15,23,22,24,17,28);(7,12)(10,11)(13,17,19,22,20,15,18,14)(16,23,27,21,25,28,26,24);(1,3,4,2)(16,25)(21,24)(23,28)(26,27)

Polycyclic presentation:

2^9|a^2=gh,b^2=fi,c^2=hi,d^2=gi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,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, 7511416);