#210651

order	512
nilpotency class	3
isoclinism	isoclinic class of #24390
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

623 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=h,d^2=gh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,c^b=ci,d^b=dh,d^c=di,f^c=fh

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, 117847);