#7185319

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

2005 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)(14,19)(16,17)(20,22)(21,26)(23,25)(27,28);(1,2)(3,4)(5,6,9,10)(7,8,12,11)(14,20)(19,22)(21,27)(26,28);(5,7)(6,11)(8,10)(9,12)(13,14,16,22,18,20,17,19)(15,21,25,28,24,27,23,26);(2,4)(6,10)(7,12)(13,15)(14,21)(16,25)(17,23)(18,24)(19,26)(20,27)(22,28);(5,8,9,11)(6,7,10,12)(13,16,18,17)(14,22,20,19)(15,23,24,25)(21,26,27,28)

Polycyclic presentation:

2^9|b^2=hi,c^2=gi,e^2=h,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	#72257084, #72257115, #72257166
maximal normal subgroup of	#72257084, #72257115, #72257166
maximal quotient of	#72265028

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7092515);