#300624

order	512
nilpotency class	3
isoclinism	isoclinic class of #264043
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

936 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ehi,c^2=fi,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,f^b=fi,g^b=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 207820);