#210667

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
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	C₂³	C₂×C₄×D₄
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

671 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:

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

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,f^a=fh,c^b=ci,d^b=di,d^c=dh,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, 117863);