#7683364

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	C₂×#537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6738 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7590560);