#7647251

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×#550	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8372 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)(5,6,9,8)(7,11,12,10)(18,26)(21,25)(23,24)(27,28);(1,2)(3,4)(13,14,19,17)(15,22,16,20)(18,21,26,25)(23,28,24,27);(5,7)(6,10)(8,11)(9,12)(13,15,19,16)(14,20,17,22)(18,23,26,24)(21,27,25,28);(6,8)(10,11)(13,16,19,15)(14,20,17,22)(18,24,26,23)(21,27,25,28);(5,8,9,6)(7,11,12,10)(13,17,19,14)(15,22,16,20)(18,25,26,21)(23,28,24,27);(5,7)(6,10)(8,11)(9,12)(13,18)(14,21)(15,23)(16,24)(17,25)(19,26)(20,27)(22,28)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=i,e^2=hi,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, 7554447);