#7654563

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

8308 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)(6,9)(7,11)(13,14,19,23)(15,21,16,20)(17,18,22,28)(24,27,26,25);(1,2,3,4)(6,9)(10,12);(5,6,8,9)(7,10,11,12)(13,15,19,16)(14,20,23,21)(17,24,22,26)(18,25,28,27);(5,7,8,11)(6,10,9,12)(13,16,19,15)(14,21,23,20)(17,26,22,24)(18,27,28,25);(13,17)(14,18)(15,24)(16,26)(19,22)(20,25)(21,27)(23,28);(13,18)(14,22)(15,25)(16,27)(17,23)(19,28)(20,26)(21,24)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=h,d^2=h,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	#72304538, #72304609, #72304625
maximal normal subgroup of	#72304538, #72304609, #72304625

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7561759);