#6371652

order	512
nilpotency class	3
isoclinism	isoclinic class of #60367
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3238 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=fhi,c^2=f,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,d^b=di,g^c=gi

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, 6278848);