#6373786

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	C₂²×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3048 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)(18,21)(22,34)(24,35)(26,31)(28,40)(29,39)(30,36)(32,37)(33,38);(9,11,21,36,14,24,13,23)(10,15,27,35,19,30,18,20)(12,22,37,40,25,38,26,39)(16,28,32,34,17,29,31,33);(1,2,3,4)(9,12)(10,16)(11,22)(13,26)(14,25)(15,28)(17,19)(18,31)(20,33)(21,37)(23,39)(24,38)(27,32)(29,30)(34,35)(36,40);(5,7,8,6)(9,12)(10,17)(11,22)(13,26)(14,25)(15,29)(16,19)(18,32)(20,34)(21,37)(23,39)(24,38)(27,31)(28,30)(33,35)(36,40);(12,25)(16,17)(22,38)(26,37)(28,29)(31,32)(33,34)(39,40)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=gi,d^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,e^c=ei,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, 6280982);