#493032

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	C₂×#525	C₂²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1300 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fi,d^2=i,b^a=be,d^a=di,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, 400228);