#7630528

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

7680 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)(6,11)(7,12)(13,14)(15,20)(16,17)(18,22)(19,23)(21,27)(24,25)(26,28);(5,6)(7,8)(9,12)(10,11)(13,15)(14,19)(16,26)(17,25)(18,23)(20,22)(21,24)(27,28);(1,2,3,4)(5,7,10,12)(6,8,11,9)(15,23)(19,20)(24,26)(25,28);(5,8,10,9)(6,7,11,12)(13,15)(14,20)(16,24)(17,25)(18,23)(19,22)(21,26)(27,28);(5,9,10,8)(6,12,11,7)(13,16,18,21)(14,17,22,27)(15,24,23,26)(19,28,20,25);(13,17)(14,21)(15,25)(16,22)(18,27)(19,24)(20,26)(23,28)

Polycyclic presentation:

2^9|c^2=h,d^2=gi,e^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7537724);