#7754838

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

8588 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)(7,8)(9,10)(11,12)(13,14)(15,19)(16,22)(17,21)(18,20)(23,26)(24,28)(25,27);(1,2)(3,4)(13,15)(14,19)(16,20)(17,23)(18,22)(21,26)(24,27)(25,28);(6,10)(7,11)(14,20)(15,22)(17,25)(24,26);(5,7,9,11)(6,8,10,12)(13,14)(15,16)(17,21)(18,20)(19,22)(23,24)(25,27)(26,28);(5,8)(6,11)(7,10)(9,12)(13,16)(14,15)(17,24)(18,19)(20,22)(21,23)(25,26)(27,28);(13,17)(14,21)(15,23)(16,24)(18,25)(19,26)(20,27)(22,28)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261985
maximal normal subgroup of	#72261985

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7662034);