#7692394

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

6938 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=h,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7599590);