#7626543

order	512
nilpotency class	2
isoclinism	isoclinic class of #62862
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁶
commutator	C₂²	C₂⁵×C₄
prime radical	C₂×#3543	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7600 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=g,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh,d^c=dh

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72297126
maximal normal subgroup of	#72297126
maximal quotient of	#72297185, #72297400, #72297978

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533739);