#7867830

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

7397 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)(13,14,18,15)(16,21,24,19)(17,20,25,22)(23,27,28,26);(1,2)(3,4)(5,7,11,9)(6,8,12,10);(5,8,11,10)(6,9,12,7)(14,15)(19,21)(20,22)(26,27);(5,7,11,9)(6,8,12,10)(13,15,18,14)(16,21,24,19)(17,22,25,20)(23,27,28,26);(5,9,11,7)(6,8,12,10)(13,16,18,24)(14,19,15,21)(17,23,25,28)(20,26,22,27);(5,10,11,8)(6,9,12,7)(13,17,18,25)(14,20,15,22)(16,23,24,28)(19,26,21,27)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238225, #72238235, #72238245
maximal normal subgroup of	#72238225, #72238235, #72238245

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7775026);