#298333

order	512
nilpotency class	3
isoclinism	isoclinic class of #269211
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	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

895 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)(7,16,18,12)(8,17,19,20)(9,15,11,13)(21,22,24,27)(23,26,31,35)(25,33,32,28)(29,34,30,36);(1,2,4,3)(5,7)(6,12)(8,15)(9,17)(10,18)(11,20)(13,19)(14,16)(21,23)(22,26)(24,31)(25,29)(27,35)(28,36)(30,32)(33,34);(5,8,9,16,10,19,11,12)(6,7,13,20,14,18,15,17)(22,27)(23,29,31,30)(26,34,35,36)(28,33);(1,3,4,2)(5,9,10,11)(6,13,14,15)(7,17,18,20)(8,12,19,16)

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=fi,d^2=e,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=dh,g^c=gh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 205529);