#7672708

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

8178 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 20 points:

(2,4)(6,10)(8,11)(14,20)(15,18);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(13,14)(15,17)(16,18)(19,20);(6,10)(7,12)(13,15,19,18)(14,16,20,17);(13,16,19,17)(14,18,20,15);(5,7,9,12)(6,11,10,8)(13,17,19,16)(14,15,20,18);(5,8,9,11)(6,12,10,7)(13,18,19,15)(14,16,20,17)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7579904);