#461182

order	512
nilpotency class	3
isoclinism	isoclinic class of #460096
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2760
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

1035 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)(17,30,32,24)(19,23)(20,28,34,26)(22,29)(31,35)(33,36);(5,7)(6,11)(8,19,21,16)(9,17)(10,18)(12,25,27,23)(13,24)(14,15)(20,33,34,31)(22,32)(26,36,28,35)(29,30);(5,8)(6,12,14,27)(7,16)(9,20)(10,21)(11,23,15,25)(13,28,29,26)(17,31)(18,19)(22,34)(24,36,30,35)(32,33);(2,4)(5,9,10,22)(6,13,14,29)(7,17,18,32)(8,20,21,34)(11,24,15,30)(12,26,27,28)(16,31,19,33)(23,35,25,36)

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,g^b=gi,d^c=de,f^c=fi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285210, #72285222, #72285268, …
maximal normal subgroup of	#72285210, #72285222, #72285268, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 368378);