#7634874

order	512
nilpotency class	2
isoclinism	isoclinic class of #36702
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5642 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

(1,2,4,6)(3,7,8,5)(9,10)(11,13)(12,14)(15,16)(17,18)(19,25)(20,24)(21,23)(22,26)(27,31)(28,30)(29,32);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16);(10,14)(11,15)(17,19)(18,22)(20,27)(21,26)(23,25)(24,29)(28,32)(30,31);(10,14)(13,16)(18,23)(19,26)(24,30)(27,32);(18,23)(22,25)(24,30)(29,31);(17,20,21,28)(18,24,23,30)(19,27,26,32)(22,29,25,31)

Polycyclic presentation:

2^9|a^2=g,b^2=g,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	#72303568, #72303570, #72303576
maximal normal subgroup of	#72303568, #72303570, #72303576
maximal quotient of	#72303616

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7542070);