#202679

order	512
nilpotency class	3
isoclinism	isoclinic class of #24390
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 ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

749 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)(7,15)(8,9)(10,13)(11,14)(12,17)(16,19)(18,20)(21,22,26,30,25,29,31,35)(23,24,28,33,32,27,34,36);(5,7,9,17,11,19,10,18)(6,12,13,15,14,20,8,16)(21,23,31,34,25,32,26,28)(22,27,35,33,29,24,30,36);(2,4)(5,8,9,6,11,13,10,14)(7,16,17,12,19,15,18,20)(21,24,25,27)(22,28,29,34)(23,30,32,35)(26,33,31,36);(5,9,11,10)(6,8,14,13)(7,17,19,18)(12,16,20,15)(23,32)(24,27)(28,34)(33,36)

Polycyclic presentation:

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

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, 109875);