#359229

order	512
nilpotency class	3
isoclinism	isoclinic class of #23400
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1224 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=ei,d^2=eh,f^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,f^b=fi,d^c=di,f^c=fi

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, 266425);