#361103

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

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	C₂²×C₄⋊C₄
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

842 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)(7,12)(8,10)(9,11)(13,14,17,24,19,25,18,23)(15,22,26,20,27,21,16,28);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,17,26,19,27,18,16)(14,20,23,22,25,28,24,21);(2,4)(5,8,9,7)(6,10,11,12)(13,16,17,15,19,26,18,27)(14,21,23,20,25,22,24,28);(6,11)(10,12)(13,16,18,27,19,26,17,15)(14,22,24,20,25,21,23,28)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268299);