#322993

order	512
nilpotency class	3
isoclinism	isoclinic class of #288931
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

718 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 230189);