#6671841

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	#23854	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2164 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(6,9)(7,11)(13,14,19,25)(15,26,30,21)(16,23,17,33)(18,24,20,35)(22,34,39,32)(27,41,28,37)(29,42,31,38)(36,43,40,44);(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,15,19,30)(14,21,25,26)(16,27,17,28)(18,29,20,31)(22,36,39,40)(23,37,33,41)(24,38,35,42)(32,43,34,44);(2,4)(13,16,19,17)(14,22,25,39)(15,27,30,28)(18,32,20,34)(21,36,26,40)(23,24,33,35)(29,43,31,44)(37,38,41,42);(5,7,8,11)(6,10,9,12)(13,17,19,16)(14,23,25,33)(15,28,30,27)(18,32,20,34)(21,37,26,41)(22,24,39,35)(29,43,31,44)(36,38,40,42);(2,4)(6,9)(10,12)(13,18,19,20)(14,24,25,35)(15,29,30,31)(16,32,17,34)(21,38,26,42)(22,33,39,23)(27,43,28,44)(36,41,40,37)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,g^c=gi,e^d=ei

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, 6579037);