#7653296

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×#3513	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7310 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)(13,14,19,22)(15,23,26,20)(16,25,27,17)(18,21,28,24);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,15,19,26)(14,20,22,23)(16,24,27,21)(17,18,25,28);(5,7)(6,9)(8,12)(10,11)(13,15,19,26)(14,20,22,23)(16,24,27,21)(17,18,25,28);(13,16)(14,17)(15,24)(18,20)(19,27)(21,26)(22,25)(23,28);(5,7)(6,10)(8,12)(9,11)(13,17)(14,16)(15,18)(19,25)(20,24)(21,23)(22,27)(26,28);(5,6)(7,9)(8,11)(10,12)(13,18)(14,21)(15,25)(16,20)(17,26)(19,28)(22,24)(23,27)

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^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, 7560492);