#301228

order	512
nilpotency class	3
isoclinism	isoclinic class of #264363
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×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

926 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 208424);