#7715936

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

7300 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)(6,10)(8,11)(13,14)(15,20)(16,18)(17,24)(19,22)(21,25)(23,26)(27,28);(1,2,3,4)(5,6,9,10)(7,11,12,8)(13,15)(14,19)(16,20)(17,23)(18,22)(21,26)(24,27)(25,28);(6,10)(7,12)(14,16)(15,22)(21,24)(23,28);(13,15)(14,20)(16,19)(17,23)(18,22)(21,27)(24,26)(25,28);(5,7)(6,8)(9,12)(10,11)(13,16,18,14)(15,19,22,20)(17,24,25,21)(23,26,28,27);(5,8)(6,7)(9,11)(10,12)(13,17)(14,21)(15,23)(16,24)(18,25)(19,26)(20,27)(22,28)

Polycyclic presentation:

2^9|b^2=gh,e^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,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, 7623132);