#7806581

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

7126 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,f^b=fi,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, 7713777);