#7633182

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

7449 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)(6,9)(10,12)(13,14,19,23)(15,20,26,27)(16,25,17,21)(18,22,28,24);(1,2,3,4)(13,15)(14,20)(16,22)(17,24)(18,21)(19,26)(23,27)(25,28);(5,6)(7,10)(8,9)(11,12)(13,16,19,17)(14,21,23,25)(15,24,26,22)(18,20,28,27);(5,7,8,11)(6,10,9,12)(13,17,19,16)(14,21,23,25)(15,24,26,22)(18,27,28,20);(13,15)(14,20)(16,24)(17,22)(18,25)(19,26)(21,28)(23,27);(13,18,19,28)(14,22,23,24)(15,25,26,21)(16,20,17,27)

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7540378);