#7726607

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

7634 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)(7,8)(9,10)(11,12)(13,14,19,20)(15,22,16,18)(17,25,26,21)(23,27,24,28);(1,2,3,4)(13,15,19,16)(14,18,20,22)(17,23,26,24)(21,27,25,28);(6,10)(7,11)(14,20)(15,16)(17,26)(27,28);(5,7,9,11)(6,8,10,12)(13,16,19,15)(14,18,20,22)(17,24,26,23)(21,27,25,28);(5,8)(6,11)(7,10)(9,12)(13,17,19,26)(14,21,20,25)(15,23,16,24)(18,27,22,28);(13,18)(14,15)(16,20)(17,27)(19,22)(21,23)(24,25)(26,28)

Polycyclic presentation:

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

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, 7633803);