#7672207

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

7100 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,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, 7579403);