#391423

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

912 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 298619);