#298382

order	512
nilpotency class	3
isoclinism	isoclinic class of #268211
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	#2819	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

939 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)(8,16,19,15)(9,17,18,13)(21,22)(23,29)(24,27)(25,34)(26,31)(28,33)(30,35)(32,36);(1,2,4,3)(5,7,10,12)(6,11,14,20)(8,17,19,13)(9,16,18,15)(21,23,24,31)(22,26,27,29)(25,35,33,32)(28,30,34,36);(5,8,18,20,10,19,9,11)(6,12,17,15,14,7,13,16)(21,22)(23,30,31,36)(24,27)(25,28)(26,35,29,32)(33,34);(1,3,4,2)(22,27)(26,29)(28,34)(30,36)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=fh,d^2=ehi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=dhi,g^c=gh

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 205578);