#7527770

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2170 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307820, #72307988, #72308281
maximal normal subgroup of	#72307820, #72307988, #72308281

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7434966);