#372798

order	512
nilpotency class	2
isoclinism	isoclinic class of #14715
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

1348 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=e,d^2=efg,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 279994);