#519201

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄×#120
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

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

2484 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 426397);