#6671967

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	#23853	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2152 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)(13,14,18,24)(15,26,30,20)(16,31,33,22)(17,25,19,23)(21,35,39,32)(27,40,28,37)(29,38,34,41)(36,44,42,43);(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,15,18,30)(14,20,24,26)(16,27,33,28)(17,34,19,29)(21,36,39,42)(22,40,31,37)(23,41,25,38)(32,44,35,43);(2,4)(13,16)(14,21)(15,27)(17,32)(18,33)(19,35)(20,36)(22,25)(23,31)(24,39)(26,42)(28,30)(29,43)(34,44)(37,41)(38,40);(5,7,8,11)(6,10,9,12)(13,16)(14,22)(15,28)(17,35)(18,33)(19,32)(20,37)(21,25)(23,39)(24,31)(26,40)(27,30)(29,43)(34,44)(36,41)(38,42);(2,4)(6,9)(10,12)(13,17,18,19)(14,23,24,25)(15,29,30,34)(16,32,33,35)(20,38,26,41)(21,31,39,22)(27,43,28,44)(36,40,42,37)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=fi,e^2=i,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,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, 6579163);