#851320

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4368 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=g,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232108, #72232136, #72232175, …
maximal normal subgroup of	#72232108, #72232136, #72232175, …
maximal quotient of	C₂×#851320, #72331302, #72336331

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 758516);