#7551156

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518992
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	C₂×#3267	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2145 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=g,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263899, #72263980, #72264032
maximal normal subgroup of	#72263899, #72263980, #72264032
maximal quotient of	#72264924

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7458352);