#6460004

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2479 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)(13,14,20,17,19,24,18,23)(15,25,16,21,27,28,26,22);(6,9)(10,12)(13,15)(14,21)(16,18)(17,25)(19,27)(20,26)(22,24)(23,28);(1,2,3,4)(13,16,18,15,19,26,20,27)(14,22,23,28,24,21,17,25);(5,7,8,11)(6,10,9,12)(13,17,18,14,19,23,20,24)(15,22,26,28,27,21,16,25);(13,18,19,20)(14,23,24,17)(15,16,27,26)(21,28,22,25)

Polycyclic presentation:

2^9|a^2=fi,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,e^c=ei

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, 6367200);