#7627241

order	512
nilpotency class	2
isoclinism	isoclinic class of #63011
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁶
commutator	C₂²	C₂⁵×C₄
prime radical	C₂×D₄×#41	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8446 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)(5,6,9,8)(7,11,12,10)(13,14)(15,20)(16,19)(17,21)(18,23)(22,26)(24,27)(25,28);(5,7)(6,10)(8,11)(9,12)(13,14)(15,20)(16,19)(17,21)(18,23)(22,26)(24,27)(25,28);(5,7)(6,10)(8,11)(9,12)(13,15)(14,18)(16,22)(17,23)(19,25)(20,21)(24,28)(26,27);(5,7)(6,10)(8,11)(9,12)(13,16,17,24)(14,19,21,27)(15,22,23,28)(18,25,20,26);(5,8,9,6)(7,11,12,10)(13,15)(14,20)(16,22)(17,23)(18,21)(19,26)(24,28)(25,27);(5,7)(6,11)(8,10)(9,12)(13,14)(15,18)(16,19)(17,21)(20,23)(22,25)(24,27)(26,28)

Polycyclic presentation:

2^9|a^2=i,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,f^e=fh

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304536, #72304577, #72304604
maximal normal subgroup of	#72304536, #72304577, #72304604

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534437);