#6345318

order	512
nilpotency class	3
isoclinism	isoclinic class of #61591
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	C₂×#449	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1607 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,4,8,5,7,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,29)(20,25)(21,27)(22,26)(23,28)(24,31)(30,43)(32,38)(33,42)(34,40)(35,41)(36,39)(37,45)(44,48)(46,47);(17,19)(18,24)(20,30)(21,31)(22,36)(23,32)(25,37)(26,29)(27,38)(28,39)(33,44)(34,45)(35,46)(40,47)(41,48)(42,43);(1,3,4,10,5,9,11,15)(2,6,8,13,7,12,14,16)(17,20)(18,25)(19,30)(21,34)(22,33)(23,35)(24,37)(26,42)(27,40)(28,41)(29,43)(31,45)(32,46)(36,44)(38,47)(39,48);(3,9)(6,12)(10,15)(13,16)(17,21,23,22)(18,26,28,27)(19,31,32,36)(20,33,35,34)(24,29,39,38)(25,40,41,42)(30,44,46,45)(37,47,48,43);(2,7)(6,12)(8,14)(13,16)(17,21,23,22)(18,27,28,26)(19,31,32,36)(20,34,35,33)(24,38,39,29)(25,40,41,42)(30,45,46,44)(37,47,48,43)

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,d^a=dh,e^a=ei,f^a=fh,f^b=fh,d^c=dhi

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, 6252514);