#7185325

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1999 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257063, #72257084, #72257181, …
maximal normal subgroup of	#72257063, #72257084, #72257181, …
maximal quotient of	#72265426

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7092521);