#460331

order	512
nilpotency class	3
isoclinism	isoclinic class of #460330
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3058	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1260 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)(5,6,10,15,11,16,9,13)(7,17,20,26,21,28,18,12)(8,22,24,30,25,29,23,14)(19,33,35,36,34,27,32,31);(5,7,9,18,11,21,10,20)(6,12,15,17,16,26,13,28)(8,19,23,32,25,34,24,35)(14,27,22,31,30,33,29,36);(6,13)(7,18)(8,23)(9,10)(12,26)(15,16)(19,34)(20,21)(22,29)(24,25)(27,36)(31,33);(2,4)(5,8,10,24,11,25,9,23)(6,14,13,29,16,30,15,22)(7,19,20,35,21,34,18,32)(12,27,28,36,26,33,17,31)

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=ei,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^b=fi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=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, 367527);