#7712679

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7378 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)(6,10)(8,11)(13,14,18,22)(15,19,26,23)(16,27,25,20)(17,28,24,21);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(13,14,18,22)(15,19,26,23)(16,20,25,27)(17,21,24,28);(6,10)(7,12)(13,14)(15,23)(16,27)(17,21)(18,22)(19,26)(20,25)(24,28);(13,15)(14,19)(16,24)(17,25)(18,26)(20,28)(21,27)(22,23);(5,7,9,12)(6,11,10,8)(13,16,18,25)(14,20,22,27)(15,24,26,17)(19,28,23,21);(5,8,9,11)(6,12,10,7)(13,17,18,24)(14,21,22,28)(15,25,26,16)(19,27,23,20)

Polycyclic presentation:

2^9|a^2=h,b^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei

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, 7619875);