#367904

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1173 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)(5,6,8,7)(9,10,15,21,16,22,14,20)(11,23,27,17,28,36,26,37)(12,29,32,39,25,35,31,18)(13,33,30,19,24,40,34,38);(1,2,4,3)(5,7,8,6)(9,11,15,26,16,28,14,27)(10,17,21,23,22,37,20,36)(12,30,32,13,25,34,31,24)(18,38,29,40,39,19,35,33);(9,12,16,25)(10,18,22,39)(11,24,28,13)(14,32,15,31)(17,33,37,40)(19,36,38,23)(20,29,21,35)(26,34,27,30);(1,3,4,2)(9,13,16,24)(10,19,22,38)(11,25,28,12)(14,30,15,34)(17,35,37,29)(18,36,39,23)(20,40,21,33)(26,31,27,32)

Polycyclic presentation:

2^9|a^2=h,b^2=egh,c^2=i,d^2=efi,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=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, 275100);