#7709076

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

7404 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)(15,19)(16,24)(17,22)(18,20)(21,26)(23,28)(25,27);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,17)(14,19,20,22)(16,23,26,25)(21,27,24,28);(6,9)(10,12)(14,20)(19,22)(21,24)(27,28);(13,14)(15,19)(16,21)(17,22)(18,20)(23,27)(24,26)(25,28);(5,6)(7,10)(8,9)(11,12)(13,16,18,26)(14,21,20,24)(15,23,17,25)(19,27,22,28);(5,7)(6,10)(8,11)(9,12)(13,17,18,15)(14,22,20,19)(16,25,26,23)(21,28,24,27)

Polycyclic presentation:

2^9|b^2=gh,e^2=h,f^2=h,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, 7616272);