#516637

order	512
nilpotency class	2
isoclinism	isoclinic class of #34980
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

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

5658 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)(5,6,7,8)(10,13)(14,16)(17,18,21,23)(19,24,22,20);(2,4)(5,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,20)(21,22)(23,24);(1,2)(3,4)(5,6)(7,8)(11,15)(14,16)(17,20,21,24)(18,22,23,19);(9,11)(10,14)(12,15)(13,16)(17,20)(18,19)(21,24)(22,23);(9,10)(11,14)(12,13)(15,16)(17,19)(18,20)(21,22)(23,24)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=gh,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^c=df

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267604, #72267900, #72267901
maximal normal subgroup of	#72267604, #72267900, #72267901
maximal quotient of	#72267612

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 423833);