#367032

order	512
nilpotency class	3
isoclinism	isoclinic class of #25140
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1470 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=efgh,c^2=i,d^2=ei,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, 274228);