#544366

order	512
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator	C₂³	C₂⁴×C₄
Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#525	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

5538 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=i,e^2=f,b^a=bf,c^a=cg,c^b=ch,d^b=dfg,d^c=df

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313578
maximal normal subgroup of	#72313578
maximal quotient of	#72313643

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 451562);