#7857059

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

8426 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)(7,10)(8,11)(9,12)(13,14,20,25)(15,19,24,28)(16,23,18,21)(17,22,26,27);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,19)(16,17)(18,26)(20,24)(21,27)(22,23)(25,28);(5,6)(7,10)(8,11)(9,12)(13,16,20,18)(14,21,25,23)(15,26,24,17)(19,22,28,27);(13,17,20,26)(14,22,25,27)(15,16,24,18)(19,23,28,21);(5,6)(7,9)(8,11)(10,12)(13,18,20,16)(14,23,25,21)(15,26,24,17)(19,22,28,27);(5,7)(6,10)(8,12)(9,11)(13,19,20,28)(14,24,25,15)(16,22,18,27)(17,23,26,21)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7764255);