#7682473

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

6911 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)(6,10)(11,12)(13,14,19,16)(15,18,22,20)(17,21,27,24)(23,26,25,28);(5,6)(7,11)(8,12)(9,10)(13,15,19,22)(14,20,16,18)(17,25,27,23)(21,26,24,28);(2,4)(15,22)(18,20)(23,25)(26,28);(1,2)(3,4)(5,7,9,8)(6,11,10,12)(13,16,19,14)(15,18,22,20)(17,21,27,24)(23,28,25,26);(5,8,9,7)(6,12,10,11)(13,17,19,27)(14,21,16,24)(15,23,22,25)(18,26,20,28);(13,18)(14,15)(16,22)(17,26)(19,20)(21,23)(24,25)(27,28)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=g,e^2=g,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, 7589669);