#8091735

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

7552 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)(5,6)(7,10)(8,9)(11,12)(13,14)(15,20)(16,21)(17,22)(18,19)(23,27)(24,25)(26,28);(1,2)(3,4)(5,7)(6,10)(8,12)(9,11);(6,8)(7,11)(14,19)(17,27)(20,25)(21,28);(7,11)(10,12)(13,15)(14,20)(16,22)(17,21)(18,24)(19,25)(23,26)(27,28);(5,8,9,6)(7,12,11,10)(13,16)(14,21)(15,23)(17,25)(18,26)(19,28)(20,27)(22,24);(5,7)(6,10)(8,12)(9,11)(13,17,18,27)(14,22,19,23)(15,21,24,28)(16,25,26,20)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238143, #72238165, #72238245
maximal normal subgroup of	#72238143, #72238165, #72238245

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7998931);