#7631844

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

8260 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)(11,12)(13,14)(15,24)(16,20)(17,21)(18,22)(19,23)(25,28)(26,27);(5,6)(7,11)(8,12)(9,10)(13,15)(14,18)(16,26)(17,25)(19,22)(20,28)(21,27)(23,24);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(13,16,19,17)(14,20,23,21)(15,25,22,26)(18,27,24,28);(5,7,9,8)(6,11,10,12)(13,17,19,16)(14,21,23,20)(15,25,22,26)(18,27,24,28);(5,8,9,7)(6,12,10,11)(13,18,19,24)(14,22,23,15)(16,27,17,28)(20,26,21,25);(13,14)(15,24)(16,20)(17,21)(18,22)(19,23)(25,28)(26,27)

Polycyclic presentation:

2^9|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, #72302972, #72302976
maximal normal subgroup of	#72302359, #72302972, #72302976

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7539040);