#6445012

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
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	#24271	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2058 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)(13,14,18,23)(15,24,27,20)(16,28,30,43)(17,22,19,33)(21,29,37,31)(25,39,41,42)(26,36,32,35)(34,40,38,44);(1,2)(3,4)(6,9)(10,12)(13,15)(14,20)(16,25)(17,32)(18,27)(19,26)(21,34)(22,36)(23,24)(28,42)(29,44)(30,41)(31,40)(33,35)(37,38)(39,43);(2,4)(13,16,19,31,18,30,17,29)(14,21,22,28,23,37,33,43)(15,25,26,40,27,41,32,44)(20,34,36,42,24,38,35,39);(2,4)(13,17,18,19)(14,22,23,33)(15,26,27,32)(16,29,30,31)(20,35,24,36)(21,28,37,43)(25,40,41,44)(34,39,38,42);(5,7,8,11)(6,10,9,12)(15,27)(16,30)(20,24)(21,37)(26,32)(28,43)(29,31)(35,36)

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,e^c=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, 6352208);