#298534

order	512
nilpotency class	3
isoclinism	isoclinic class of #269339
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	#10228	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1119 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)(5,6,10,12)(7,8,11,9)(13,14,18,22)(15,24,26,20)(16,25,27,28)(17,23,19,21);(1,2,4,3)(5,7)(6,8)(9,12)(10,11)(13,15)(14,20)(16,23)(17,25)(18,26)(19,28)(21,27)(22,24);(5,7,8,12,10,11,9,6)(13,16,19,20,18,27,17,24)(14,15,23,25,22,26,21,28);(1,3,4,2)(5,8,10,9)(6,11,12,7)(13,17,18,19)(14,21,22,23)(15,25,26,28)(16,20,27,24)

Polycyclic presentation:

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

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, 205730);