#7633245

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	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7914 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)(5,6,8,10)(7,11,12,9)(13,14,18,22)(15,16,20,19)(17,21,26,25)(23,24,28,27);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14,18,22)(15,16,20,19)(17,25,26,21)(23,27,28,24);(5,7,8,12)(6,9,10,11)(13,15)(14,19)(16,22)(17,23)(18,20)(21,27)(24,25)(26,28);(13,15)(14,16)(17,23)(18,20)(19,22)(21,24)(25,27)(26,28);(13,16)(14,20)(15,22)(17,24)(18,19)(21,28)(23,25)(26,27);(13,17,18,26)(14,21,22,25)(15,23,20,28)(16,24,19,27)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=hi,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, 7540441);