#7837858

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

7058 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=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, 7745054);