#394471

order	512
nilpotency class	3
isoclinism	isoclinic class of #394470
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#393
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2108 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,d^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,g^b=gi,e^c=ei

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, 301667);