#7199131

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996607
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 ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1939 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308113, #72308768
maximal normal subgroup of	#72308113, #72308768

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7106327);