#7660629

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×#3520	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7023 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)(13,14,19,23)(15,16,25,20)(17,21,24,26)(18,22,27,28);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12);(5,7)(6,9)(8,12)(10,11)(13,15,19,25)(14,20,23,16)(17,18,24,27)(21,28,26,22);(5,6)(7,10)(8,11)(9,12)(13,16,19,20)(14,15,23,25)(17,22,24,28)(18,26,27,21);(5,6)(7,9)(8,11)(10,12)(13,17)(14,21)(15,18)(16,22)(19,24)(20,28)(23,26)(25,27);(5,7)(6,10)(8,12)(9,11)(13,18)(14,22)(15,24)(16,26)(17,25)(19,27)(20,21)(23,28)

Polycyclic presentation:

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

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, 7567825);