#7676065

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6910 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=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, 7583261);