#7198621

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2011	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1762 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(2,4)(5,6,10,7)(8,11,15,13)(9,14,17,12)(16,19,20,18)(21,22,27,25,28,30,26,29)(23,31,33,32,34,36,35,24);(5,7,10,6)(8,11,15,13)(9,12,17,14)(16,19,20,18)(21,23)(22,24)(25,36)(26,33)(27,35)(28,34)(29,31)(30,32);(1,2,3,4)(5,8,10,15)(6,11,7,13)(9,16,17,20)(12,18,14,19)(21,24)(22,23)(25,33)(26,36)(27,31)(28,32)(29,35)(30,34);(8,15)(11,13)(16,20)(18,19)(21,25,27,30,28,29,26,22)(23,32,33,36,34,24,35,31);(5,9,10,17)(6,12,7,14)(8,16,15,20)(11,18,13,19)(21,26,28,27)(22,29,30,25)(23,33,34,35)(24,31,32,36)

Polycyclic presentation:

2^9|a^2=fh,b^2=h,c^2=gh,d^2=f,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257127, #72257134, #72257144
maximal normal subgroup of	#72257127, #72257134, #72257144
maximal quotient of	#72308495

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7105817);