#282536

order	512
nilpotency class	3
isoclinism	isoclinic class of #281915
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#25599	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

847 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=f,c^2=hi,d^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=deh,c^b=cg,d^b=dh,g^b=gi,d^c=di,f^c=fh,g^c=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 189732);