#7744843

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

8234 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)(13,14)(15,21)(16,17)(18,22)(19,25)(20,27)(23,24)(26,28);(1,2)(3,4)(5,6)(7,11)(8,9)(10,12)(13,14)(15,21)(16,17)(18,22)(19,25)(20,27)(23,24)(26,28);(6,9)(10,11)(13,15,18,25)(14,19,22,21)(16,26,20,23)(17,24,27,28);(13,16,18,20)(14,17,22,27)(15,23,25,26)(19,28,21,24);(5,7)(6,10)(8,12)(9,11)(13,17)(14,20)(15,24)(16,22)(18,27)(19,23)(21,26)(25,28);(5,6)(7,10)(8,9)(11,12)(13,15)(14,21)(16,23)(17,24)(18,25)(19,22)(20,26)(27,28)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72302966, #72302976, #72303013
maximal normal subgroup of	#72302966, #72302976, #72303013

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7652039);