#351849

order	512
nilpotency class	3
isoclinism	isoclinic class of #348358
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
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

705 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)(5,6,10,12)(7,14,16,20)(8,17,19,11)(9,18,15,13)(21,22)(23,29)(24,26)(25,33)(27,31)(28,32)(30,36)(34,35);(5,7,15,17,10,16,9,11)(6,8,18,20,12,19,13,14)(22,26)(27,35)(28,33)(29,36);(1,2,3,4)(5,8,10,19)(6,11,12,17)(7,13,16,18)(9,14,15,20)(22,26)(27,35)(28,33)(29,36);(6,12)(8,19)(13,18)(14,20)(21,23,24,30)(22,27,26,35)(25,34,32,31)(28,36,33,29)

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=fi,d^2=i,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gi,c^b=ch,d^c=dh,g^d=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 259045);