#366601

order	512
nilpotency class	3
isoclinism	isoclinic class of #25140
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2252 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)(11,15)(12,23)(13,27)(14,18)(16,25)(17,19)(20,36)(21,34)(22,32)(24,39)(26,35)(28,31)(29,33)(30,37)(38,40);(2,4)(9,11,19,21,14,22,13,28)(10,15,17,34,18,32,27,31)(12,24,29,38,26,20,25,37)(16,30,23,39,33,40,35,36);(6,8)(9,12,14,26)(10,16,18,33)(11,20,22,24)(13,29,19,25)(15,30,32,40)(17,35,27,23)(21,38,28,37)(31,39,34,36);(2,4)

Polycyclic presentation:

2^9|a^2=h,b^2=fghi,c^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,f^c=fi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 273797);