#298243

order	512
nilpotency class	3
isoclinism	isoclinic class of #271131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1062 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:

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

Polycyclic presentation:

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

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, 205439);