#358747

order	512
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂²×C₄⋊C₄
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

986 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,d^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg

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, 265943);