#433541

order	512
nilpotency class	3
isoclinism	isoclinic class of #427374
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2760
commutator, Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

906 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)(8,12,21,27)(9,13)(10,14)(16,25)(17,24,31,28)(19,23)(20,26,33,36)(22,29)(30,34)(32,35);(5,7)(6,11)(8,19,21,16)(9,17)(10,18)(12,25,27,23)(13,28)(14,15)(20,32,33,30)(22,31)(24,29)(26,34,36,35);(5,8)(6,12,14,27)(7,16)(9,20)(10,21)(11,23,15,25)(13,26,29,36)(17,30)(18,19)(22,33)(24,34,28,35)(31,32);(2,4)(5,9,10,22)(6,13,14,29)(7,17,18,31)(8,20,21,33)(11,24,15,28)(12,26,27,36)(16,30,19,32)(23,34,25,35)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285210, #72285332, #72285355, …
maximal normal subgroup of	#72285210, #72285332, #72285355, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 340737);