#6443920

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23421	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2139 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)(7,12)(8,10)(9,11)(13,14)(15,19)(16,24)(17,23)(18,21)(20,25)(22,28)(26,27);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,18,17)(14,19,21,23)(16,25,26,22)(20,27,28,24);(2,4)(13,16,15,22,18,26,17,25)(14,20,23,27,21,28,19,24);(2,4)(5,8,9,7)(6,10,11,12)(13,14)(15,23)(16,20)(17,19)(18,21)(22,27)(24,25)(26,28);(13,15,18,17)(14,19,21,23)(16,22,26,25)(20,24,28,27)

Polycyclic presentation:

2^9|b^2=fi,c^2=gi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,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, 6351116);