#374441

order	512
nilpotency class	3
isoclinism	isoclinic class of #27987
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

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	C₂×#3058	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1730 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fhi,d^2=ei,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,f^c=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, 281637);