#378783

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

864 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317237, #72317285, #72317288
maximal normal subgroup of	#72317237, #72317285, #72317288

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 285979);