#7654888

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×#3481	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7284 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)(13,14)(15,22)(16,21)(17,20)(18,23)(19,26)(24,27)(25,28);(1,2,3,4)(5,6)(7,12)(8,10)(9,11)(13,15)(14,19)(16,24)(17,25)(18,26)(20,27)(21,28)(22,23);(5,7,9,10)(6,8,11,12)(13,16,18,17)(14,20,23,21)(15,24,26,25)(19,27,22,28);(13,17,18,16)(14,21,23,20)(15,25,26,24)(19,28,22,27);(5,6)(7,8)(9,11)(10,12)(13,15)(14,22)(16,24)(17,25)(18,26)(19,23)(20,28)(21,27);(5,8)(6,10)(7,11)(9,12)(13,14)(15,19)(16,20)(17,21)(18,23)(22,26)(24,27)(25,28)

Polycyclic presentation:

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

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, 7562084);