#515327

order	512
nilpotency class	2
isoclinism	isoclinic class of #35346
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

6702 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252215, #72252216, #72252228
maximal normal subgroup of	#72252215, #72252216, #72252228

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 422523);