C₂×#25588 #6449080

order	512
nilpotency class	3
isoclinism	isoclinic class of #24464
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4740 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 18 points:

(1,2);(3,4)(5,9)(6,7)(8,10)(13,18)(15,17);(4,7)(8,9)(11,12)(13,15)(14,16)(17,18);(12,15,16,17);(3,5,6,10)(4,8,7,9)

maximal subgroup of	#72319215, #72319417, #72338522, …
maximal normal subgroup of	#72319215, #72319417, #72338522, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6356276);