#7523584

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35750	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2393 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257502, #72257577, #72264317
maximal normal subgroup of	#72257502, #72257577, #72264317
maximal quotient of	#72307875

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7430780);