#392827

order	512
nilpotency class	3
isoclinism	isoclinic class of #32947
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

1471 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)(9,21)(12,25)(13,23)(14,24)(16,31)(17,30)(18,19)(27,28)(29,32)(33,36);(1,3,4,2)(5,6,9,11,10,14,21,24)(7,12,15,22,17,25,30,34)(8,13,18,23,20,28,19,27)(16,26,29,33,32,35,31,36);(5,7,9,15,10,17,21,30)(6,12,11,22,14,25,24,34)(8,16,18,29,20,32,19,31)(13,26,23,33,28,35,27,36);(5,8)(6,13)(7,16)(9,18)(10,20)(11,23)(12,26)(14,28)(15,29)(17,32)(19,21)(22,33)(24,27)(25,35)(30,31)(34,36)

Polycyclic presentation:

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

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, 300023);