#454971

order	512
nilpotency class	3
isoclinism	isoclinic class of #454970
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23402	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1554 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:

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

Polycyclic presentation:

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

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, 362167);