#455075

order	512
nilpotency class	3
isoclinism	isoclinic class of #454110
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23407	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1493 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:

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

Polycyclic presentation:

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

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, 362271);