#7732553

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

7111 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)(15,21)(17,25)(19,23)(20,24);(1,2)(3,4)(5,6,9,7)(8,11,12,10)(13,14,18,16)(15,19,21,23)(17,20,25,24)(22,26,27,28);(6,7)(10,11)(13,15)(14,19)(16,23)(17,22)(18,21)(20,26)(24,28)(25,27);(13,16,18,14)(15,19,21,23)(17,24,25,20)(22,26,27,28);(5,7,9,6)(8,11,12,10)(13,17)(14,20)(15,22)(16,24)(18,25)(19,26)(21,27)(23,28);(5,8,9,12)(6,10,7,11)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7639749);