#155753

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁵	C₂⁴
commutator	C₂⁴	C₂³×C₄
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

3842 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=egh,c^2=g,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72293955, #72293965, #72293981, …
maximal normal subgroup of	#72293955, #72293965, #72293981, …
maximal quotient of	C₂×#155753

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 62949);