#372508

order	512
nilpotency class	3
isoclinism	isoclinic class of #196038
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#95 ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1617 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=e,d^2=fg,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,h^c=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, 279704);