#7812175

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,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, 7719371);