#7625445

order	512
nilpotency class	2
isoclinism	isoclinic class of #3484
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator	C₂²	C₂⁵×C₄
Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8374 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,f^2=h,b^a=bg,c^a=ch,e^a=eh,d^c=dh

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315552, #72315590
maximal normal subgroup of	#72315552, #72315590
maximal quotient of	#72316293

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532641);