#6445173

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2032 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6352369);