#371300

order	512
nilpotency class	3
isoclinism	isoclinic class of #24464
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1399 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)(14,18,20,24)(19,28,21,27);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(13,14)(15,21)(16,24)(17,20)(18,26)(19,23)(22,28)(25,27);(6,9)(10,12)(13,15,17,23)(14,19,20,21)(16,25,26,22)(18,27,24,28);(5,7,8,11)(6,10,9,12)(13,14)(15,19)(16,24)(17,20)(18,26)(21,23)(22,27)(25,28)

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=i,d^2=gi,e^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,e^d=ei,f^d=fi,h^d=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, 278496);