#7631033

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
prime radical	C₂×#3485	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8846 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)(6,9)(7,11)(13,14)(15,19)(16,24)(17,21)(18,23)(20,25)(22,26)(27,28);(5,6)(7,10)(8,9)(11,12)(13,15,17,23)(14,18,21,19)(16,22,25,28)(20,26,24,27);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,14,17,21)(15,19,23,18)(16,20,25,24)(22,27,28,26);(13,14)(15,18)(16,20)(17,21)(19,23)(22,26)(24,25)(27,28);(13,15)(14,19)(16,22)(17,23)(18,21)(20,27)(24,26)(25,28);(13,16)(14,20)(15,22)(17,25)(18,26)(19,27)(21,24)(23,28)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313631
maximal normal subgroup of	#72313631

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538229);