#455944

order	512
nilpotency class	3
isoclinism	isoclinic class of #455561
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1159 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)(3,4)(5,6,11,17)(7,18,21,12)(8,14,24,32)(9,16,10,15)(13,26,31,23)(19,30,34,36)(20,27,35,28)(22,29,25,33);(5,7,9,25,11,21,10,22)(6,12,15,33,17,18,16,29)(8,20,23,34,24,35,26,19)(13,30,14,28,31,36,32,27);(5,8)(6,13)(7,19,21,34)(9,26)(10,23)(11,24)(12,27,18,28)(14,16)(15,32)(17,31)(20,25,35,22)(29,30,33,36);(2,4)(5,8)(6,14)(7,20)(9,23)(10,26)(11,24)(12,28)(13,16)(15,31)(17,32)(18,27)(19,22)(21,35)(25,34)(29,30)(33,36)

Polycyclic presentation:

2^9|a^2=i,b^2=ef,c^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,d^c=de,f^c=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, 363140);