#492996

order	512
nilpotency class	4
isoclinism	isoclinic class of #33348
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1257 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400192);