#7806598

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

7922 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72236848, #72238144, #72238191
maximal normal subgroup of	#72236848, #72238144, #72238191
maximal quotient of	#72263468, #72263484

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7713794);