#149467

order	512
nilpotency class	4
isoclinism	isoclinic class of #13307
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

710 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,d^2=hi,e^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,d^b=dh,h^b=hi,e^c=ei,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56663);