#460132

order	512
nilpotency class	3
isoclinism	isoclinic class of #460131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2760
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

1349 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)(7,11,17,25)(8,12)(9,21)(10,14)(13,22)(15,29,18,23)(16,31,33,24)(19,26)(20,27)(28,34)(30,36,32,35);(1,2)(3,4)(5,7)(6,11,14,25)(8,18)(9,16)(10,17)(12,23,27,29)(13,24,21,31)(15,20)(19,32)(22,33)(26,35,28,36)(30,34);(2,4)(5,8)(6,12)(7,15,17,18)(9,19)(10,20)(11,23,25,29)(13,28)(14,27)(16,32,33,30)(21,26)(22,34)(24,35,31,36);(5,9,10,22)(6,13)(7,16,17,33)(8,19,20,34)(11,24)(12,26)(14,21)(15,30,18,32)(23,35)(25,31)(27,28)(29,36)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283557, #72283657, #72283682, …
maximal normal subgroup of	#72283557, #72283657, #72283682, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 367328);