#202353

order	512
nilpotency class	3
isoclinism	isoclinic class of #32878
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	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

784 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)(7,15)(8,14)(9,11)(10,13)(12,16)(17,20)(18,19)(21,22,24,28,25,27,31,34)(23,26,29,33,30,32,35,36);(1,2,4,3)(6,11)(12,19)(13,14)(15,20)(21,23,24,29,25,30,31,35)(22,26,28,33,27,32,34,36);(1,3,4,2)(5,7,9,17)(6,12,11,19)(8,16,10,18)(13,15,14,20);(5,8,9,10)(6,13,11,14)(7,16,17,18)(12,15,19,20)(22,27)(26,32)(28,34)(33,36)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=e,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,f^a=fh,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, 109549);