#357407

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator	C₂³	C₂²×C₄²
socle, prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1114 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=eg,d^2=ef,g^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cef,d^b=dei

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, 264603);