#7631859

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

8142 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)(5,6,9,12)(7,10,8,11)(13,14,20,24)(15,25,27,21)(16,18,17,22)(19,23,28,26);(5,6)(7,10)(8,11)(9,12)(13,15)(14,21)(16,23)(17,26)(18,19)(20,27)(22,28)(24,25);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,16,20,17)(14,22,24,18)(15,26,27,23)(19,25,28,21);(5,7,9,8)(6,10,12,11)(13,17,20,16)(14,22,24,18)(15,26,27,23)(19,21,28,25);(5,8,9,7)(6,11,12,10)(13,18,20,22)(14,17,24,16)(15,19,27,28)(21,26,25,23);(13,19)(14,23)(15,22)(16,25)(17,21)(18,27)(20,28)(24,26)

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72302359, #72302985, #72302991
maximal normal subgroup of	#72302359, #72302985, #72302991

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7539055);