#7744990

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

7302 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:

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

Polycyclic presentation:

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

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, 7652186);