#7688413

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

8304 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)(7,9,12,11)(13,14)(15,23)(16,17)(18,21)(19,22)(20,27)(24,26)(25,28);(1,2)(3,4)(13,15,19,21)(14,18,22,23)(16,25,27,24)(17,26,20,28);(5,6)(7,11)(8,10)(9,12)(13,16,19,27)(14,20,22,17)(15,24,21,25)(18,26,23,28);(5,7)(6,9)(8,12)(10,11)(13,16)(14,17)(15,25)(18,26)(19,27)(20,22)(21,24)(23,28);(13,17)(14,16)(15,26)(18,25)(19,20)(21,28)(22,27)(23,24);(13,18)(14,21)(15,22)(16,26)(17,24)(19,23)(20,25)(27,28)

Polycyclic presentation:

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

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7595609);