#7650147

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×#3511	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6970 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=h,e^2=hi,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, 7557343);