#7626255

order	512
nilpotency class	2
isoclinism	isoclinic class of #62817
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×#3485	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8486 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72302286, #72302931, #72302952
maximal normal subgroup of	#72302286, #72302931, #72302952

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533451);