#205456

order	512
nilpotency class	3
isoclinism	isoclinic class of #19456
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1022 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=fh,d^2=h,e^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh,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, 112652);