#367541

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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

930 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)(11,21)(12,26)(13,20)(14,17)(15,19)(16,24)(18,27)(22,31)(23,37)(25,34)(28,33)(29,32)(30,39)(35,38)(36,40);(1,2,3,4)(5,7,8,6)(9,11,13,24,14,25,15,22)(10,16,19,21,17,31,20,34)(12,23,27,38,28,39,29,36)(18,30,26,35,32,37,33,40);(2,4)(10,17)(11,22,25,24)(16,21,31,34)(18,32)(19,20)(23,36,39,38)(26,33)(30,35,37,40);(9,12,13,27,14,28,15,29)(10,18,19,26,17,32,20,33)(11,23,24,38,25,39,22,36)(16,30,21,35,31,37,34,40)

Polycyclic presentation:

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

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, 274737);