#205043

order	512
nilpotency class	3
isoclinism	isoclinic class of #163983
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35303	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1217 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=f,d^2=h,e^2=hi,g^2=hi,b^a=be,c^a=cf,e^a=eh,e^b=ei,g^b=gi,d^c=dhi

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, 112239);