#6466143

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

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

2244 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6373339);