#7683261

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

8519 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72302317, #72302988, #72302991
maximal normal subgroup of	#72302317, #72302988, #72302991

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7590457);