#495039

order	512
nilpotency class	4
isoclinism	isoclinic class of #33618
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

632 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)(6,12)(7,16)(8,10)(9,17)(13,26)(14,29)(15,21)(18,30)(19,32)(20,31)(22,23)(24,28)(27,35)(33,36);(1,3,4,2)(5,6)(7,13)(8,21)(9,14)(10,12)(11,15)(16,28)(17,25)(18,27)(19,33)(20,24)(22,34)(23,29)(26,32)(30,36)(31,35);(5,7,17,32,10,20,23,30,11,18,22,31,8,19,9,16)(6,13,25,26,12,24,29,36,15,27,34,35,21,33,14,28);(7,18)(13,27)(16,30)(19,20)(24,33)(26,35)(28,36)(31,32)

Polycyclic presentation:

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

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, 402235);