#7711974

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

7260 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)(7,11)(10,12)(13,14,19,23)(15,24,20,16)(17,21,22,18)(25,26,28,27);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,20)(14,16,23,24)(17,25,22,28)(18,26,21,27);(6,9)(10,12)(13,16,19,24)(14,20,23,15)(17,26,22,27)(18,28,21,25);(13,17,19,22)(14,21,23,18)(15,25,20,28)(16,27,24,26);(5,6)(7,10)(8,9)(11,12)(13,14)(15,16)(17,21)(18,22)(19,23)(20,24)(25,27)(26,28);(5,7,8,11)(6,10,9,12)(13,18,19,21)(14,22,23,17)(15,26,20,27)(16,28,24,25)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=h,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, 7619170);