#7642327

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

7274 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7549523);