#359228

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,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, 266424);