#7199189

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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×#3306	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1970 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)(5,6)(7,10)(8,11)(9,12)(13,14,20,17,19,23,18,24)(15,25,28,21,16,22,26,27);(5,6)(7,10)(8,11)(9,12)(13,15)(14,21)(16,19)(17,25)(18,28)(20,26)(22,24)(23,27);(1,2,3,4)(5,7,10,6)(8,11,9,12)(13,16,19,15)(14,22,23,25)(17,27,24,21)(18,28,20,26);(5,8,10,9)(6,11,7,12)(13,17,18,14,19,24,20,23)(15,21,26,25,16,27,28,22);(5,9,10,8)(6,11,7,12)(14,23)(15,16)(17,24)(26,28)

Polycyclic presentation:

2^9|a^2=fi,c^2=gi,d^2=fhi,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^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307857, #72307988, #72308294
maximal normal subgroup of	#72307857, #72307988, #72308294

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7106385);