#367380

order	512
nilpotency class	3
isoclinism	isoclinic class of #219498
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1066 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226670, #72226672, #72226675
maximal normal subgroup of	#72226670, #72226672, #72226675

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 274576);