#456595

order	512
nilpotency class	3
isoclinism	isoclinic class of #454369
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23399	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1396 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)(7,17,21,32)(8,13)(9,14,18,33)(10,15)(11,16)(12,20,30,22)(19,34)(23,31)(24,29)(25,28,26,35)(27,36);(1,2)(3,4)(5,7,10,22,11,21,27,20)(6,12,15,32,16,30,36,17)(8,18,23,25,24,9,19,26)(13,28,31,14,29,35,34,33);(2,4)(5,8,11,24)(6,13,16,29)(7,18,21,9)(10,19,27,23)(12,28,30,35)(14,32,33,17)(15,34,36,31)(20,25,22,26);(5,9)(6,14)(7,19)(8,22)(10,26)(11,18)(12,29)(13,30)(15,35)(16,33)(17,31)(20,24)(21,23)(25,27)(28,36)(32,34)

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,h^c=hi,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, 363791);