#517942

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, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2514 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=f,d^2=i,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, 425138);