#127006

order	512
nilpotency class	4
isoclinism	isoclinic class of #11541
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂×#117	C₂³
socle	C₂²	C₂×#342
prime radical	C₂⁵	C₄²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

605 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,e^2=h,f^2=i,b^a=bd,d^a=dg,g^a=gi,c^b=chi,e^b=eg,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34202);