#262228

order	512
nilpotency class	3
isoclinism	isoclinic class of #246847
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

859 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)(5,6,10,14)(7,15,17,8)(9,20,11,13)(12,19,16,18)(21,22)(23,29)(24,32)(25,28)(26,31)(27,33)(30,36)(34,35);(1,2,4,3)(5,7,10,17)(6,8,14,15)(9,16,11,12)(13,18,20,19)(24,33)(26,35)(27,32)(31,34);(5,8,10,15)(6,12,14,16)(7,13,17,20)(9,19,11,18)(21,23,25,30)(22,26,28,35)(24,34,33,31)(27,36,32,29);(1,3,4,2)(5,9,10,11)(6,13,14,20)(7,16,17,12)(8,18,15,19)(23,30)(26,35)(29,36)(31,34)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 169424);