#6466541

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
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

2014 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)(13,14,18,21)(15,22,20,17)(16,19,24,26)(23,28,27,25);(2,3)(6,9)(10,11)(13,15,17,14,18,20,22,21)(16,23,25,19,24,27,28,26);(1,2,4,3)(5,7,8,12)(6,10,9,11)(13,16)(14,19)(15,23)(17,25)(18,24)(20,27)(21,26)(22,28);(1,3,4,2);(16,24)(19,26)(23,27)(25,28)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=gh,d^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^c=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, 6373737);