#361652

order	512
nilpotency class	3
isoclinism	isoclinic class of #25798
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3584	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3532 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281024, #72281136, #72281219, …
maximal normal subgroup of	#72281024, #72281136, #72281219, …
maximal quotient of	#72313306

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268848);