#289645

order	512
nilpotency class	3
isoclinism	isoclinic class of #281915
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

750 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=efi,d^2=ei,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,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, 196841);