#278720

order	512
nilpotency class	3
isoclinism	isoclinic class of #268763
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	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

688 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)(13,14,19,27,18,23,17,22)(15,20,24,28,16,26,25,21);(1,2,4,3)(5,6,8,11)(7,9,12,10)(13,15)(14,20)(16,18)(17,25)(19,24)(21,22)(23,26)(27,28);(6,9)(7,12)(10,11)(13,16,18,15)(14,21,23,28)(17,25,19,24)(20,27,26,22);(1,3,4,2)(13,17,18,19)(14,22,23,27)(15,24,16,25)(20,28,26,21)

Polycyclic presentation:

2^9|a^2=fh,b^2=ei,c^2=h,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,g^b=gi,f^c=fh,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 185916);