#6459959

order	512
nilpotency class	3
isoclinism	isoclinic class of #24264
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2462 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi

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, 6367155);