#456893

order	512
nilpotency class	3
isoclinism	isoclinic class of #453991
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1181 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:

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

Polycyclic presentation:

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

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, 364089);