#456505

order	512
nilpotency class	3
isoclinism	isoclinic class of #456505
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23400	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1203 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,h^b=hi,d^c=de,e^c=ei,h^c=hi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 363701);