#7842485

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

6947 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)(5,6,9,10)(7,8,11,12)(13,14,18,23)(15,19,25,20)(16,26,27,21)(17,22,28,24);(1,2,3,4)(13,15)(14,19)(16,24)(17,26)(18,25)(20,23)(21,28)(22,27);(6,10)(7,11)(13,15,18,25)(14,20,23,19)(16,24,27,22)(17,26,28,21);(5,7)(6,8)(9,11)(10,12)(13,15)(14,19)(16,22)(17,21)(18,25)(20,23)(24,27)(26,28);(5,8)(6,7)(9,12)(10,11)(13,16)(14,21)(15,24)(17,20)(18,27)(19,28)(22,25)(23,26);(13,17,18,28)(14,22,23,24)(15,21,25,26)(16,20,27,19)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^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, 7749681);