#298065

order	512
nilpotency class	3
isoclinism	isoclinic class of #271259
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

1028 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)(7,13)(8,17)(9,12)(10,19)(11,15)(14,20)(16,18)(21,22)(23,26)(24,27)(25,33)(28,32)(29,34)(30,36)(31,35);(1,2,4,3)(5,7)(6,11)(8,18)(9,15)(10,16)(12,13)(14,19)(17,20)(21,23)(22,26)(24,31)(25,29)(27,35)(28,36)(30,32)(33,34);(5,8,17,6,9,19,10,12)(7,14,20,13,15,18,16,11)(22,27)(23,29,31,30)(26,34,35,36)(28,33);(1,3,4,2)(6,12)(7,15)(8,19)(16,20)

Polycyclic presentation:

2^9|b^2=eh,c^2=fhi,d^2=eh,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=dh

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, 205261);