#7648716

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×#3505	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7182 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)(13,14)(15,20)(16,26)(17,18)(19,23)(21,25)(22,24)(27,28);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,25)(14,20,23,21)(16,17,24,27)(18,22,28,26);(6,9)(10,12)(13,15,19,25)(14,21,23,20)(16,17,24,27)(18,26,28,22);(7,11)(10,12)(13,16)(14,22)(15,17)(18,21)(19,24)(20,28)(23,26)(25,27);(5,7)(6,10)(8,11)(9,12)(13,17)(14,18)(15,24)(16,25)(19,27)(20,22)(21,26)(23,28);(5,6)(7,10)(8,9)(11,12)(13,18)(14,17)(15,22)(16,21)(19,28)(20,24)(23,27)(25,26)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7555912);