#374757

order	512
nilpotency class	3
isoclinism	isoclinic class of #25440
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1070 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)(9,10,14,19,15,20,13,16)(11,17,23,29,24,33,21,32)(12,25,27,34,28,30,26,18)(22,36,37,31,38,39,35,40);(1,2)(3,4)(10,16,20,19)(11,21,24,23)(18,30,34,25)(22,35,38,37);(2,4)(5,7,8,6)(9,11,13,23,15,24,14,21)(10,17,19,32,20,33,16,29)(12,22,26,37,28,38,27,35)(18,31,25,36,34,40,30,39);(9,12,14,27,15,28,13,26)(10,18,16,30,20,34,19,25)(11,22,21,35,24,38,23,37)(17,31,29,39,33,40,32,36)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 281953);