#7633156

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

8244 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)(5,6,9,12)(7,8,11,10)(13,14,19,23)(15,20,25,28)(16,26,24,21)(17,22,27,18);(5,7,9,11)(6,10,12,8)(13,15)(14,20)(16,27)(17,24)(18,26)(19,25)(21,22)(23,28);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,16,19,24)(14,21,23,26)(15,17,25,27)(18,28,22,20);(5,8,9,10)(6,11,12,7)(13,15)(14,20)(16,17)(18,21)(19,25)(22,26)(23,28)(24,27);(13,17)(14,22)(15,24)(16,25)(18,23)(19,27)(20,21)(26,28);(13,18,19,22)(14,17,23,27)(15,21,25,26)(16,28,24,20)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=i,d^2=g,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, 7540352);