#412343

order	512
nilpotency class	3
isoclinism	isoclinic class of #411585
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1262 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 319539);