#367382

order	512
nilpotency class	3
isoclinism	isoclinic class of #27987
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1002 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=ef,d^2=e,e^2=i,f^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,g^c=gi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 274578);