#7641990

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×#3444	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7236 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)(8,12)(10,11)(13,14,17,20)(15,21,23,18)(16,19,24,26)(22,27,28,25);(1,2,3,4)(6,7)(10,11)(16,24)(19,26)(22,28)(25,27);(5,6,9,7)(8,10,12,11)(13,14)(15,18)(16,19)(17,20)(21,23)(22,25)(24,26)(27,28);(13,15,17,23)(14,18,20,21)(16,22,24,28)(19,25,26,27);(5,7,9,6)(8,11,12,10)(13,16,17,24)(14,19,20,26)(15,22,23,28)(18,25,21,27);(5,8)(6,10)(7,11)(9,12)(13,14)(15,18)(16,19)(17,20)(21,23)(22,25)(24,26)(27,28)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,d^2=h,e^2=hi,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^b=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, 7549186);