#379822

order	512
nilpotency class	3
isoclinism	isoclinic class of #214688
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35676	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2146 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)(7,16,19,11)(8,13)(9,12)(10,14)(15,22)(17,30,32,25)(18,26,33,24)(20,28)(21,29)(23,27)(31,35,34,36);(1,2)(3,4)(5,7)(6,11)(8,17)(9,18)(10,19)(12,24)(13,25)(14,16)(15,26)(20,31)(21,32)(22,33)(23,34)(27,35)(28,36)(29,30);(2,4)(6,12)(9,22)(11,24)(13,28)(14,15)(16,26)(18,33)(20,23)(25,36)(27,29)(30,35)(31,34);(5,8,10,21)(6,13,14,29)(7,17,19,32)(9,23,22,20)(11,25,16,30)(12,27,15,28)(18,34,33,31)(24,35,26,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318090, #72318152, #72318255
maximal normal subgroup of	#72318090, #72318152, #72318255

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 287018);