#7712005

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

7218 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)(13,14,18,19)(15,20,26,23)(16,27,25,21)(17,28,24,22);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(14,19)(20,23)(21,27)(22,28);(6,9)(10,12)(13,14)(15,23)(16,27)(17,22)(18,19)(20,26)(21,25)(24,28);(13,15)(14,20)(16,24)(17,25)(18,26)(19,23)(21,28)(22,27);(5,6)(7,10)(8,9)(11,12)(13,16,18,25)(14,21,19,27)(15,24,26,17)(20,28,23,22);(5,7,8,11)(6,10,9,12)(13,17,18,24)(14,22,19,28)(15,25,26,16)(20,27,23,21)

Polycyclic presentation:

2^9|a^2=h,e^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, 7619201);