#7641747

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8437 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)(15,21)(16,20)(17,19)(18,22)(23,27)(24,26)(25,28);(2,4)(7,12)(9,11)(13,14)(15,18)(16,20)(17,19)(21,22)(23,25)(24,26)(27,28);(5,7,8,12)(6,9,10,11)(13,15,17,22)(14,18,19,21)(16,23,24,28)(20,25,26,27);(6,10)(9,11)(14,19)(18,21)(20,26)(25,27);(15,22)(18,21)(23,28)(25,27);(13,16,17,24)(14,20,19,26)(15,23,22,28)(18,25,21,27)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280398, #72282531, #72302332, …
maximal normal subgroup of	#72280398, #72282531, #72302332, …
maximal quotient of	#72281124, #72281384, #72283690

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7548943);