#7604355

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518952
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 ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1945 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)(13,14,19,21)(15,25,26,23)(16,24,18,22)(17,20,27,28);(2,3)(5,7)(6,10)(8,11)(9,12)(13,15,16,27,19,26,18,17)(14,20,22,23,21,28,24,25);(1,2,4,3)(5,8,9,6)(7,11,12,10)(14,21)(15,26)(17,27)(22,24);(1,3,4,2)(7,12)(10,11)(13,16,19,18)(14,22,21,24)(15,27,26,17)(20,23,28,25);(13,17,18,26,19,27,16,15)(14,23,24,20,21,25,22,28)

Polycyclic presentation:

2^9|a^2=gh,b^2=fi,c^2=h,d^2=g,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,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, 7511551);