#6387646

order	512
nilpotency class	3
isoclinism	isoclinic class of #61629
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	C₂×#3002	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2778 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=dfi,e^b=ei,g^c=gi

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, 6294842);