#518018

order	512
nilpotency class	2
isoclinism	isoclinic class of #35680
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2190 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=hi,e^2=fg,b^a=bf,c^a=cg,c^b=ch,e^d=ef

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 425214);