#361698

order	512
nilpotency class	3
isoclinism	isoclinic class of #24468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3499	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3600 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)(9,10,15,18,14,20,13,19)(11,21,23,16,25,33,24,32)(12,26,28,17,30,36,29,35)(22,31,39,37,27,34,38,40);(5,6,7,8)(9,11,14,25)(10,16,20,32)(12,27,30,22)(13,24,15,23)(17,34,35,31)(18,33,19,21)(26,37,36,40)(28,38,29,39);(2,4)(6,8)(9,12,14,30)(10,17,20,35)(11,22,25,27)(13,28,15,29)(16,31,32,34)(18,26,19,36)(21,37,33,40)(23,38,24,39);(9,11)(10,16)(12,22)(13,24)(14,25)(15,23)(17,31)(18,33)(19,21)(20,32)(26,40)(27,30)(28,39)(29,38)(34,35)(36,37)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,e^b=ei,f^c=fi,h^c=hi,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, 268894);