#392861

order	512
nilpotency class	3
isoclinism	isoclinic class of #32950
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1445 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gi,f^c=fi,g^d=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300057);