#397475

order	512
nilpotency class	3
isoclinism	isoclinic class of #395148
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1278 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=i,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 304671);