#298263

order	512
nilpotency class	3
isoclinism	isoclinic class of #271131
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	#39491	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1096 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)(5,6,10,7)(8,11,12,9)(13,14)(15,17)(16,23)(18,22)(19,25)(20,26)(21,27)(24,28);(1,2,4,3)(5,7)(6,10)(8,11)(9,12)(13,15,17,14)(16,21,24,19)(18,22,26,20)(23,25,28,27);(5,8,9,6,10,12,11,7)(13,16,18,25,17,24,26,27)(14,19,20,23,15,21,22,28);(1,3,4,2)(5,9,10,11)(6,12,7,8)(14,15)(16,24)(20,22)(25,27)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 205459);