#7653994

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

8438 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)(15,23)(16,18)(17,20)(19,22)(21,27)(24,28)(25,26);(1,2)(3,4)(7,12)(9,11);(5,7,8,12)(6,9,10,11)(13,15,19,17)(14,20,22,23)(16,24,27,26)(18,25,21,28);(5,6)(7,9)(8,10)(11,12)(13,16)(14,21)(15,24)(17,26)(18,22)(19,27)(20,28)(23,25);(13,17,19,15)(14,20,22,23)(16,26,27,24)(18,25,21,28);(13,18,19,21)(14,16,22,27)(15,25,17,28)(20,24,23,26)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304561, #72304599, #72304611
maximal normal subgroup of	#72304561, #72304599, #72304611

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7561190);