#7645702

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂²×#548	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8064 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7552898);