#432171

order	512
nilpotency class	3
isoclinism	isoclinic class of #419459
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	C₂⁵	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1544 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)(7,11,17,23)(8,12)(9,13)(10,14)(15,28,30,21)(16,22,31,26)(18,24)(19,25)(20,27)(29,36,32,34)(33,35);(5,7)(6,11,14,23)(8,15,19,30)(9,16,20,31)(10,17)(12,21)(13,26)(18,32)(22,27)(24,34,35,36)(25,28)(29,33);(5,8,10,19)(6,12,14,25)(7,15)(9,18,20,33)(11,21)(13,24,27,35)(16,29)(17,30)(22,34)(23,28)(26,36)(31,32);(2,4)(5,9)(6,13)(7,16,17,31)(8,18)(10,20)(11,22,23,26)(12,24)(14,27)(15,29,30,32)(19,33)(21,34,28,36)(25,35)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285122, #72285210, #72285400, …
maximal normal subgroup of	#72285122, #72285210, #72285400, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 339367);