#7199169

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2048 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)(13,14,18,23,19,17,22,28)(15,21,24,27,16,25,26,20);(5,6)(7,10)(8,11)(9,12)(13,15)(14,20)(16,19)(17,27)(18,26)(21,28)(22,24)(23,25);(1,2,3,4)(5,7,10,6)(8,11,9,12)(13,16,19,15)(14,21,17,25)(18,26,22,24)(20,23,27,28);(5,8,10,9)(6,11,7,12)(13,17,18,28,19,14,22,23)(15,21,24,27,16,25,26,20);(5,9,10,8)(6,11,7,12)(14,17)(15,16)(23,28)(24,26)

Polycyclic presentation:

2^9|a^2=f,c^2=g,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	#72307505, #72307710, #72308135
maximal normal subgroup of	#72307505, #72307710, #72308135

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7106365);