#392824

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

1447 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)(9,20)(12,24)(13,22)(14,23)(18,31)(21,26)(25,32)(27,33)(30,36)(34,35);(1,3,4,2)(5,6,9,11,10,14,20,23)(7,12,17,21,15,26,29,24)(8,13,18,22,19,27,31,33)(16,25,30,32,28,35,36,34);(5,7,9,17,10,15,20,29)(6,12,11,21,14,26,23,24)(8,16,18,30,19,28,31,36)(13,25,22,32,27,35,33,34);(5,8)(6,13)(7,16)(9,18)(10,19)(11,22)(12,25)(14,27)(15,28)(17,30)(20,31)(21,32)(23,33)(24,34)(26,35)(29,36)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=e,e^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^c=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, 300020);