#202428

order	512
nilpotency class	3
isoclinism	isoclinic class of #24378
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

787 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 109624);