#366700

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2227 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=fghi,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,e^b=ei,g^c=gi,e^d=ei

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, 273896);