#493701

order	512
nilpotency class	4
isoclinism	isoclinic class of #480268
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1208 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=i,d^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,g^c=gh,g^f=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400897);