#496433

order	512
nilpotency class	4
isoclinism	isoclinic class of #33758
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

786 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403629);