#7685122

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

7346 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)(6,7)(10,11)(13,14)(15,16)(17,20)(18,21)(19,22)(23,24)(25,27)(26,28);(5,6,9,7)(8,10,12,11)(13,14)(15,16)(17,20)(18,21)(19,22)(23,24)(25,27)(26,28);(1,2,3,4)(5,7,9,6)(8,10,12,11)(13,15,19,17)(14,20,22,16)(18,23,26,25)(21,27,28,24);(1,2)(3,4)(5,8,9,12)(6,10,7,11)(13,16,19,20)(14,15,22,17)(18,24,26,27)(21,23,28,25);(13,17,19,15)(14,20,22,16)(18,23,26,25)(21,24,28,27);(13,18,19,26)(14,21,22,28)(15,23,17,25)(16,24,20,27)

Polycyclic presentation:

2^9|b^2=g,c^2=h,d^2=gi,e^2=i,f^2=i,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, 7592318);