#7933391

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

8379 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)(7,11)(8,10)(9,12)(13,14,19,21)(15,18,24,20)(16,26,27,22)(17,28,25,23);(1,2)(3,4)(13,15,19,24)(14,20,21,18)(16,25,27,17)(22,23,26,28);(5,6,8,10)(7,9,12,11)(14,21)(15,24)(16,27)(23,28);(5,6)(7,9)(8,10)(11,12)(13,16)(14,22)(15,25)(17,24)(18,28)(19,27)(20,23)(21,26);(5,7)(6,9)(8,12)(10,11)(13,17,19,25)(14,23,21,28)(15,16,24,27)(18,22,20,26);(13,18,19,20)(14,24,21,15)(16,28,27,23)(17,26,25,22)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256671, #72256676, #72256698
maximal normal subgroup of	#72256671, #72256676, #72256698

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7840587);