#424603

order	512
nilpotency class	3
isoclinism	isoclinic class of #417538
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1404 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)(7,15,18,11)(8,12)(9,13)(10,14)(16,23,19,29)(17,24,31,27)(20,25)(21,26)(22,28)(30,35,32,34)(33,36);(5,7)(6,11)(8,19)(9,17)(10,18)(12,23)(13,27)(14,15)(16,21)(20,32)(22,31)(24,28)(25,35)(26,29)(30,33)(34,36);(5,8)(6,12)(7,16,18,19)(9,20)(10,21)(11,23,15,29)(13,25)(14,26)(17,30,31,32)(22,33)(24,34,27,35)(28,36);(2,4)(5,9,10,22)(6,13,14,28)(7,17,18,31)(8,20,21,33)(11,24,15,27)(12,25,26,36)(16,30,19,32)(23,34,29,35)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285210, #72285421, #72285495
maximal normal subgroup of	#72285210, #72285421, #72285495

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 331799);