#7630817

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

7862 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)(6,10)(7,12)(13,14)(15,16)(17,25)(18,22)(19,20)(21,26)(23,27)(24,28);(5,6,9,10)(7,11,12,8)(13,15)(14,19)(16,22)(17,23)(18,20)(21,27)(24,25)(26,28);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12)(13,16,18,19)(14,20,22,15)(17,24,26,27)(21,28,25,23);(13,15)(14,16)(17,23)(18,20)(19,22)(21,24)(25,27)(26,28);(13,14)(15,19)(16,20)(17,21)(18,22)(23,27)(24,28)(25,26);(5,8,9,11)(6,7,10,12)(13,17,18,26)(14,21,22,25)(15,23,20,28)(16,24,19,27)

Polycyclic presentation:

2^9|b^2=gi,c^2=i,f^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^d=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, 7538013);