#7652421

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

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

8050 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)(5,6,8,10)(7,11,12,9)(13,14,18,16)(15,19,21,22)(17,20,25,24)(23,26,27,28);(1,2)(3,4)(7,12)(9,11)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,28)(25,27);(5,7,8,12)(6,9,10,11)(15,21)(19,22)(23,27)(26,28);(5,6)(7,9)(8,10)(11,12)(13,16,18,14)(15,22,21,19)(17,20,25,24)(23,26,27,28);(13,17,18,25)(14,20,16,24)(15,23,21,27)(19,26,22,28);(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,28)(25,27)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=hi,e^2=hi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256682, #72256691, #72256700
maximal normal subgroup of	#72256682, #72256691, #72256700

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7559617);