#456641

order	512
nilpotency class	3
isoclinism	isoclinic class of #455826
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1376 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)(7,16)(8,12)(9,13,24,31)(10,17)(11,15)(14,27)(18,22,35,33)(19,25)(20,28,36,34)(21,29)(23,32,26,30);(1,2)(3,4)(5,7,10,12,11,19,27,29)(6,8,14,16,15,21,17,25)(9,18,26,28,24,35,23,34)(13,20,32,33,31,36,30,22);(2,4)(5,8,11,21)(6,12,15,29)(7,17,19,14)(9,22,24,33)(10,25,27,16)(13,18,31,35)(20,26,36,23)(28,32,34,30);(5,9)(6,13)(7,18)(8,20)(10,26)(11,24)(12,28)(14,32)(15,31)(16,33)(17,30)(19,35)(21,36)(22,25)(23,27)(29,34)

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,e^a=ei,g^a=gi,h^a=hi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,e^d=ei,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, 363837);