#7650569

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

7247 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)(13,14,19,24)(15,20,27,21)(16,26,25,22)(17,23,28,18);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,27)(14,20,24,21)(16,17,25,28)(18,26,23,22);(6,9)(10,12)(13,15,19,27)(14,21,24,20)(16,17,25,28)(18,22,23,26);(7,11)(10,12)(13,16)(14,22)(15,17)(18,20)(19,25)(21,23)(24,26)(27,28);(5,7,8,11)(6,10,9,12)(13,17)(14,23)(15,25)(16,27)(18,24)(19,28)(20,22)(21,26);(5,6)(7,10)(8,9)(11,12)(13,18,19,23)(14,17,24,28)(15,26,27,22)(16,20,25,21)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,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, 7557765);