#282924

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

722 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)(7,14)(8,10)(9,11)(12,15)(13,17)(16,20)(18,19)(21,22,24,27)(23,29,31,26)(25,28,32,35)(30,36,33,34);(1,3,4,2)(6,10)(8,17)(11,13)(12,14)(15,18)(19,20)(23,30)(25,32)(26,34)(28,35)(29,36)(31,33);(2,3)(6,11)(10,13)(12,19)(14,20)(21,23)(22,26)(24,31)(25,33)(27,29)(28,36)(30,32)(34,35);(5,7,9,16)(6,12,11,19)(8,18,17,15)(10,14,13,20)(23,31)(26,29)(30,33)(34,36)

Polycyclic presentation:

2^9|a^2=fh,b^2=fhi,d^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, 190120);