#6373778

order	512
nilpotency class	3
isoclinism	isoclinic class of #60457
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3010 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=g,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,e^d=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6280974);