#353288

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator	C₂³	C₂²×C₄²
socle	C₂⁴	C₂³×C₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4578 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)(7,8)(13,14,16,19,17,20,23,26)(15,18,22,25,21,24,27,28);(2,4)(9,10,11,12);(6,8)(13,15)(14,18)(16,22)(17,21)(19,25)(20,24)(23,27)(26,28);(15,21)(18,24)(22,27)(25,28)

Polycyclic presentation:

2^9|a^2=g,b^2=h,g^2=i,b^a=be,c^a=cf,d^c=di

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318516, #72318562, #72318680
maximal normal subgroup of	#72318516, #72318562, #72318680

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260484);