#6382886

order	512
nilpotency class	3
isoclinism	isoclinic class of #61058
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2014 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6290082);