#7683349

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

8124 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)(6,9)(10,11)(13,14)(15,20)(16,22)(17,18)(19,21)(23,28)(24,27)(25,26);(1,2)(3,4)(13,15,19,16)(14,20,21,22)(17,26,23,24)(18,25,28,27);(5,6)(7,11)(8,9)(10,12)(14,21)(15,16)(18,28)(24,26);(5,7,8,12)(6,10,9,11)(13,16,19,15)(14,22,21,20)(17,24,23,26)(18,27,28,25);(13,17,19,23)(14,18,21,28)(15,24,16,26)(20,27,22,25);(13,18)(14,23)(15,25)(16,27)(17,21)(19,28)(20,24)(22,26)

Polycyclic presentation:

2^9|b^2=i,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, 7590545);