#376189

order	512
nilpotency class	3
isoclinism	isoclinic class of #24295
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2008	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1581 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72311981, #72312212, #72312221
maximal normal subgroup of	#72311981, #72312212, #72312221

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283385);