#7604167

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518904
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	#2011	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1821 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)(13,14,19,23)(15,18,22,26)(16,24,27,21)(17,20,28,25);(1,2,4,3)(5,7)(6,10)(8,11)(9,12)(13,15)(14,18)(16,25)(17,24)(19,22)(20,27)(21,28)(23,26);(5,8,9,6)(7,11,12,10)(13,16,26,28,19,27,18,17)(14,20,22,21,23,25,15,24);(7,12)(10,11)(13,17,18,27,19,28,26,16)(14,21,15,20,23,24,22,25);(1,3,4,2)(13,18,19,26)(14,22,23,15)(16,17,27,28)(20,21,25,24)

Polycyclic presentation:

2^9|a^2=gh,b^2=f,c^2=hi,d^2=g,e^2=fi,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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7511363);