#7712025

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

7726 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)(7,11)(10,12)(13,14)(15,17)(16,22)(18,26)(19,20)(21,23)(24,27)(25,28);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(14,20)(15,23)(22,25)(24,26);(6,9)(10,12)(13,15)(14,21)(16,26)(17,20)(18,25)(19,23)(22,27)(24,28);(13,16)(14,22)(15,24)(17,27)(18,21)(19,28)(20,25)(23,26);(5,6)(7,10)(8,9)(11,12)(13,17,19,21)(14,23,20,15)(16,27,28,18)(22,26,25,24);(5,7,8,11)(6,10,9,12)(13,18,19,27)(14,24,20,26)(15,25,23,22)(16,21,28,17)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7619221);