#7671588

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

7184 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7578784);