#519476

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	C₂⁶	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3408 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,d^2=ghi,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, 426672);