#6373788

order	512
nilpotency class	3
isoclinism	isoclinic class of #60460
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	C₂²×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3034 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)(13,27)(14,19)(15,23)(16,25)(18,28)(21,36)(22,34)(24,35)(26,32)(29,39)(30,40)(31,38)(33,37);(9,11,13,23,14,24,28,40)(10,15,18,20,19,30,27,35)(12,22,26,39,25,21,37,38)(16,31,32,36,17,29,33,34);(1,2,3,4)(9,12)(10,16)(11,21)(13,26)(14,25)(15,29)(17,19)(18,32)(20,34)(22,24)(23,38)(27,33)(28,37)(30,31)(35,36)(39,40);(5,7,8,6)(9,12)(10,17)(11,22)(13,26)(14,25)(15,29)(16,19)(18,33)(20,34)(21,24)(23,39)(27,32)(28,37)(30,31)(35,36)(38,40);(12,25)(16,17)(21,22)(26,37)(29,31)(32,33)(34,36)(38,39)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6280984);