#378876

order	512
nilpotency class	3
isoclinism	isoclinic class of #24264
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 286072);