#7630849

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
prime radical	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6716 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=i,e^2=i,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, 7538045);