#7500918

order	512
nilpotency class	3
isoclinism	isoclinic class of #30496
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2669
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂×#2669
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3111 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=df,f^b=fi,h^b=hi,g^c=gi,h^c=hi,f^d=fi,g^d=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72213889, #72214082, #72214632, …
maximal normal subgroup of	#72213889, #72214082, #72214632, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7408114);