#7849680

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

8536 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)(5,6,9,11)(7,8,12,10)(13,14)(15,17)(16,20)(18,22)(19,21)(23,25)(24,27)(26,28);(1,2)(3,4)(13,15,19,16)(14,20,21,17)(18,24,26,23)(22,25,28,27);(5,7,9,12)(6,10,11,8)(14,21)(15,16)(18,26)(25,27);(5,7)(6,8)(9,12)(10,11)(13,16,19,15)(14,20,21,17)(18,24,26,23)(22,27,28,25);(5,8)(6,7)(9,10)(11,12)(13,17,19,20)(14,16,21,15)(18,25,26,27)(22,24,28,23);(13,18,19,26)(14,22,21,28)(15,23,16,24)(17,25,20,27)

Polycyclic presentation:

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

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, 7756876);