#371326

order	512
nilpotency class	3
isoclinism	isoclinic class of #24404
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1632 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)(13,14)(15,21,17,19)(16,24)(18,22)(20,26)(23,28,25,27);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(13,15,18,17)(14,19,22,21)(16,23,26,25)(20,27,24,28);(6,9)(10,12)(13,16)(14,20)(15,23)(17,25)(18,26)(19,27)(21,28)(22,24);(5,7,8,11)(6,10,9,12)(13,17,18,15)(14,21,22,19)(16,25,26,23)(20,28,24,27)

Polycyclic presentation:

2^9|a^2=h,b^2=fh,d^2=gi,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,h^b=hi,e^c=ei,f^d=fi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 278522);