#6393480

order	512
nilpotency class	3
isoclinism	isoclinic class of #62283
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

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×#525	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3190 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6300676);