#180299

order	512
nilpotency class	3
isoclinism	isoclinic class of #180005
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#53347	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

744 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(2,6)(3,4)(7,8)(9,10)(11,16)(12,17)(13,15)(14,18)(19,22)(20,24)(21,23)(25,26,30,28)(27,33,35,40)(29,37,39,32)(31,38,34,36);(3,4)(6,8)(10,15)(12,13)(16,22)(17,18)(19,21)(23,24)(25,27)(26,31)(28,34)(29,38)(30,35)(32,40)(33,37)(36,39);(1,2,5,7)(3,6,4,8)(9,11,14,20)(10,16,18,24)(12,21,13,19)(15,22,17,23)(25,28,30,26)(27,34,35,31)(29,32,39,37)(33,38,40,36);(1,3,5,4)(2,6,7,8)(9,12,14,13)(10,17,18,15)(11,19,20,21)(16,22,24,23)(26,28)(27,35)(32,37)(36,38)

Polycyclic presentation:

2^9|a^2=hi,c^2=gi,d^2=gh,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fi,d^b=dgi,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 87495);