#298208

order	512
nilpotency class	3
isoclinism	isoclinic class of #271131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

949 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)(5,6,10,14)(7,11,12,20)(8,17,18,16)(9,15,19,13)(21,22)(23,32)(24,35)(25,31)(26,30)(27,29)(28,34)(33,36);(1,2,4,3)(5,7)(6,11)(8,15)(9,16)(10,12)(13,18)(14,20)(17,19)(21,23,26,34)(22,28,30,32)(24,27,33,25)(29,35,31,36);(5,8,9,11,10,18,19,20)(6,12,13,17,14,7,15,16)(21,24,27,34,26,33,25,23)(22,28,31,35,30,32,29,36);(1,3,4,2)(7,12)(8,18)(11,20)(16,17)(21,25,26,27)(22,29,30,31)(23,33,34,24)(28,36,32,35)

Polycyclic presentation:

2^9|a^2=hi,b^2=e,c^2=f,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,d^c=dhi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 205404);