#6380871

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2074 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=i,e^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ci,d^b=df,f^b=fi,f^d=fi

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, 6288067);