#7678840

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

7108 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)(7,10)(8,12)(9,11)(13,14)(15,20)(16,18)(17,22)(19,21)(23,27)(24,26)(25,28);(1,2,3,4)(5,7,9,8)(6,10,11,12)(13,15,19,16)(14,20,21,18)(17,24,25,23)(22,26,28,27);(6,11)(10,12)(14,21)(15,16)(17,25)(26,27);(5,6)(7,10)(8,12)(9,11)(13,16,19,15)(14,18,21,20)(17,24,25,23)(22,26,28,27);(5,8,9,7)(6,12,11,10)(13,17)(14,22)(15,23)(16,24)(18,26)(19,25)(20,27)(21,28);(13,18)(14,15)(16,21)(17,26)(19,20)(22,23)(24,28)(25,27)

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7586036);