#7676148

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

7442 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)(6,10)(11,12)(13,14)(15,21)(16,17)(18,20)(19,23)(22,27)(24,25)(26,28);(5,6)(7,11)(8,12)(9,10)(13,15)(14,19)(16,26)(17,25)(18,23)(20,21)(22,24)(27,28);(1,2,3,4)(5,7,9,8)(6,11,10,12)(14,20)(15,23)(16,22)(25,28);(1,2)(3,4)(5,7,9,8)(6,11,10,12)(13,15)(14,21)(16,24)(17,25)(18,23)(19,20)(22,26)(27,28);(5,8,9,7)(6,12,10,11)(13,16,18,22)(14,17,20,27)(15,24,23,26)(19,28,21,25);(13,17)(14,22)(15,25)(16,20)(18,27)(19,24)(21,26)(23,28)

Polycyclic presentation:

2^9|c^2=gh,d^2=gi,e^2=g,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, 7583344);