#6676443

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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	#30711	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1868 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6583639);