#515365

order	512
nilpotency class	2
isoclinism	isoclinic class of #35346
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹¹

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

7458 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=fh,d^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	#72315862, #72315889, #72315912
maximal normal subgroup of	#72315862, #72315889, #72315912

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 422561);