#208627

order	512
nilpotency class	3
isoclinism	isoclinic class of #163983
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2856	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

845 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,c^2=i,d^2=fhi,e^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,e^b=ei,g^b=gi,d^c=dhi

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, 115823);