#6460060

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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 ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2318 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:

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

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,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, 6367256);