#6460026

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
prime radical	#23386	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2316 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)(13,14,18,23,19,17,22,27)(15,24,25,21,26,28,16,20);(6,9)(10,12)(13,15,19,26)(14,20,17,21)(16,22,25,18)(23,28,27,24);(1,2,3,4)(13,16,18,15,19,25,22,26)(14,21,23,28,17,20,27,24);(5,7,8,11)(6,10,9,12)(13,17,18,27,19,14,22,23)(15,24,25,21,26,28,16,20);(14,17)(20,21)(23,27)(24,28)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=ghi,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,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, 6367222);