#7634903

order	512
nilpotency class	2
isoclinism	isoclinic class of #36702
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

8492 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)(8,9)(10,12)(13,14)(15,22)(16,25)(17,20)(18,21)(19,24)(23,27)(26,28);(2,4)(5,6)(7,10)(8,9)(11,12);(6,9)(10,11)(13,15)(14,19)(16,26)(17,23)(18,24)(20,28)(21,22)(25,27);(5,7,8,12)(6,10,9,11)(13,16,18,17)(14,20,21,25)(15,23,24,26)(19,27,22,28);(14,21)(19,22)(20,25)(27,28);(13,17,18,16)(14,20,21,25)(15,23,24,26)(19,28,22,27)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282540, #72315590
maximal normal subgroup of	#72282540, #72315590
maximal quotient of	#72283031

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7542099);