#365043

order	512
nilpotency class	3
isoclinism	isoclinic class of #28182
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3058	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1732 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72255889, #72255892, #72255919
maximal normal subgroup of	#72255889, #72255892, #72255919

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 272239);