#6394655

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	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ 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)(5,6)(7,11)(8,10)(9,12)(13,14)(15,20)(16,22)(17,23)(18,32)(19,24)(21,28)(25,34)(26,35)(27,37)(29,38)(30,36)(31,39)(33,41)(40,43)(42,44);(1,2,4,3)(5,7)(6,9)(8,12)(10,11)(13,15,19,27)(14,20,24,37)(16,25,29,42)(17,30,18,26)(21,33,39,43)(22,34,38,44)(23,36,32,35)(28,41,31,40);(1,3,4,2)(13,16)(14,21)(15,25)(17,28)(18,31)(19,29)(20,33)(22,32)(23,38)(24,39)(26,40)(27,42)(30,41)(34,35)(36,44)(37,43);(7,12)(9,11)(13,16)(14,22)(15,25)(17,31)(18,28)(19,29)(20,34)(21,32)(23,39)(24,38)(26,41)(27,42)(30,40)(33,35)(36,43)(37,44);(6,10)(9,11)(13,17,19,18)(14,23,24,32)(15,26,27,30)(16,28,29,31)(20,35,37,36)(21,38,39,22)(25,40,42,41)(33,34,43,44)

Polycyclic presentation:

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

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, 6301851);