#282847

order	512
nilpotency class	3
isoclinism	isoclinic class of #249463
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#555	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

755 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 190043);